💡Okay, so I need to prove that a convex 13-gon cannot be divided into parallelograms. Hmm, let's see. First, I should recall what a convex polygon is. A convex polygon is one where all the interior angles are less than 180 degrees, and every side is such that a line segment between any two vertices stays inside the polygon. Got it.Now, a 13-gon is a polygon with 13 sides. Since 13 is an odd number, that might be important. I remember that when dealing with parallelograms, they have opposite sides that are both equal and parallel. So, if I'm trying to divide a polygon into parallelograms, each parallelogram would need to have pairs of parallel sides.Wait, so if I'm dividing a 13-gon into parallelograms, each parallelogram would take up some sides of the original polygon. But since 13 is odd, maybe there's a problem with pairing the sides? Let me think about that.In a polygon, each side can potentially be paired with another side if they are parallel. For a polygon to be divided into parallelograms, each side must be part of a pair of parallel sides. But in a 13-gon, since there are an odd number of sides, it's impossible to pair all sides perfectly. There will always be one side left without a parallel pair. That seems like a problem because a parallelogram requires two pairs of parallel sides.Let me try to visualize this. Imagine a convex 13-gon. If I start drawing lines inside it to form parallelograms, each parallelogram would need to have two pairs of parallel sides. But with 13 sides, I can't evenly distribute the sides into pairs. There will always be one side that doesn't have a matching parallel side. This leftover side would prevent the entire polygon from being divided into parallelograms.Is there a more formal way to express this? Maybe using graph theory or something? I'm not sure. But intuitively, the odd number of sides seems to create an imbalance that makes it impossible to pair all sides into parallelograms.Wait, maybe I can think about the angles. In a convex polygon, all interior angles are less than 180 degrees. If I try to fit parallelograms inside, each parallelogram has angles that are supplementary. But with 13 sides, the angles might not align in a way that allows for perfect tiling with parallelograms.Another thought: perhaps considering the dual graph of the polygon. If the polygon were divided into parallelograms, the dual graph would have certain properties, like being bipartite or something. But I'm not too familiar with dual graphs, so maybe that's not the best approach.Going back to the pairing idea. If I have 13 sides, and each parallelogram requires two pairs of parallel sides, then the total number of sides used in the division would have to be even, right? Because each parallelogram contributes four sides, but shared sides are counted twice. Wait, no, actually, when you divide a polygon into smaller shapes, the internal sides are shared between two shapes. So, the total number of sides in the division would be more than 13, but the original polygon has 13 sides.Hmm, maybe I'm complicating things. Let's try a simpler approach. If a convex polygon can be divided into parallelograms, then it must be possible to pair each side with another parallel side. But with 13 sides, which is odd, there must be at least one side that doesn't have a parallel counterpart. Therefore, it's impossible to divide the polygon into parallelograms.That seems to make sense. So, the key point is the odd number of sides, which prevents all sides from being paired into parallel sides necessary for forming parallelograms. I think that's the main idea here.Maybe I should also consider specific examples. For instance, a convex quadrilateral can be divided into one parallelogram if it's a parallelogram itself. A convex hexagon, which has an even number of sides, might be divisible into parallelograms, but a 13-gon, being odd, cannot. This reinforces the idea that the parity of the number of sides plays a crucial role.In summary, the odd number of sides in a convex 13-gon makes it impossible to pair all sides into parallel pairs required for dividing the polygon into parallelograms. Therefore, a convex 13-gon cannot be divided into parallelograms.

💡Okay, so I have this logarithm problem to solve: evaluate log base 1/4 of 16. Hmm, logarithms can be tricky sometimes, especially when the base is a fraction. Let me think about how to approach this.First, I remember that a logarithm log_b(a) asks the question: "To what power do we need to raise b to get a?" So in this case, log_(1/4)(16) is asking: "What exponent do we need to apply to 1/4 to get 16?" That makes sense.I also recall that logarithms with fractional bases can be converted to exponential form to make them easier to handle. So, if I let x be the value of log_(1/4)(16), then I can write this as an equation:(1/4)^x = 16Now, I need to solve for x. Hmm, (1/4) is the same as 4 to the power of -1, right? Because 1/4 is the reciprocal of 4, and reciprocals can be expressed with negative exponents. So, I can rewrite (1/4) as 4^(-1). Let me substitute that into the equation:(4^(-1))^x = 16When you raise a power to another power, you multiply the exponents. So, 4^(-1 * x) equals 16. Simplifying that, I get:4^(-x) = 16Now, I need to figure out what exponent on 4 gives me 16. I know that 4 squared is 16 because 4 times 4 is 16. So, 4^2 = 16. That means:4^(-x) = 4^2Since the bases are the same and the equation holds, the exponents must be equal. Therefore, I can set the exponents equal to each other:-x = 2Solving for x, I just need to multiply both sides by -1:x = -2So, putting it all together, log base 1/4 of 16 is equal to -2. Let me double-check my steps to make sure I didn't make a mistake.1. I started by letting x be the logarithm in question.2. I converted the logarithmic equation to its exponential form.3. I expressed 1/4 as 4^(-1) to make the bases the same.4. I applied the power rule for exponents, multiplying -1 and x.5. I recognized that 16 is 4 squared, so I set the exponents equal.6. I solved for x by multiplying both sides by -1.Everything seems to check out. I don't think I made any errors in my calculations. It's interesting because the base is a fraction, so the logarithm turned out to be a negative number, which makes sense because raising a fraction to a negative exponent gives a larger number.Another way to think about it is using the change of base formula. The change of base formula states that log_b(a) can be rewritten as log_c(a) / log_c(b) for any positive number c, not equal to 1. If I choose c to be 4, then:log_(1/4)(16) = log_4(16) / log_4(1/4)I know that log_4(16) is 2 because 4 squared is 16. And log_4(1/4) is log_4(4^(-1)), which is -1 because 4 to the power of -1 is 1/4. So, putting that together:log_(1/4)(16) = 2 / (-1) = -2This confirms my earlier result. It's good to see that both methods lead to the same answer. I feel more confident now that -2 is indeed the correct value for log base 1/4 of 16.I also remember that logarithms with bases between 0 and 1 are decreasing functions, which means as the input increases, the output decreases. In this case, since 16 is greater than 1, the logarithm should be negative, which aligns with our result of -2. If the result were positive, that would mean that (1/4) raised to a positive power equals 16, which isn't possible because raising a fraction less than 1 to a positive power makes it smaller, not larger.To visualize this, if I consider the graph of y = log_(1/4)(x), it should be a decreasing curve. When x is 1, log_(1/4)(1) is 0 because any number to the power of 0 is 1. As x increases beyond 1, the logarithm becomes negative, which is consistent with our result when x is 16.Additionally, I can think about the relationship between exponents and logarithms. Since logarithms are the inverse operations of exponentials, if I have y = b^x, then x = log_b(y). In this problem, we're given y = 16 and b = 1/4, so we're solving for x in the equation (1/4)^x = 16. As we saw earlier, x turns out to be -2.It's also helpful to remember that log_b(a) = 1 / log_a(b). So, log_(1/4)(16) is the reciprocal of log_16(1/4). Let's see what log_16(1/4) is. Since 16 is 4 squared, and 1/4 is 4 to the power of -1, we can write 1/4 as 16^(-1/2) because 16^(1/2) is 4, and the reciprocal would be 16^(-1/2). Therefore, log_16(1/4) = -1/2, and the reciprocal of that is -2, which matches our previous result.This reciprocal relationship is a useful property of logarithms and can sometimes provide an alternative method to solve problems, especially when dealing with reciprocal bases or arguments.Another thing I can consider is using natural logarithms or common logarithms to solve the problem, just to see if I get the same result. Using the change of base formula again, log_(1/4)(16) can be expressed as ln(16) / ln(1/4). Calculating this numerically:ln(16) is approximately 2.7726, and ln(1/4) is approximately -1.3863. Dividing these gives 2.7726 / (-1.3863) ≈ -2. So, even using natural logarithms, I still get -2 as the result.This numerical verification adds another layer of confidence that my answer is correct. It's always good practice to check your work using different methods, especially when dealing with mathematical concepts that can be counterintuitive, like logarithms with fractional bases.I also think about the general properties of logarithms. For example, log_b(a) = - log_(1/b)(a). In this case, log_(1/4)(16) should be equal to - log_4(16). Since log_4(16) is 2, then log_(1/4)(16) should be -2. This aligns perfectly with what I found earlier.This property is a specific case of the more general logarithm identity that log_b(a) = 1 / log_a(b). When the base is the reciprocal, it introduces a negative sign, which is consistent with our result.Furthermore, I can think about the behavior of logarithmic functions. Since the base 1/4 is less than 1, the function log_(1/4)(x) is decreasing. This means that as x increases, the value of the logarithm decreases. For example, log_(1/4)(1) is 0, log_(1/4)(4) is -1, and log_(1/4)(16) is -2. This pattern shows that each time x is multiplied by 4, the logarithm decreases by 1. That makes sense because (1/4)^(-1) is 4, (1/4)^(-2) is 16, and so on.Understanding the behavior of the function helps me see why the logarithm of a larger number with a base less than 1 results in a more negative value. It's a good way to intuitively grasp why the answer is negative.I also recall that logarithms can be expressed in terms of other logarithms with different bases, which is the essence of the change of base formula. This is particularly useful when the base isn't convenient or when I don't have a calculator that can compute logarithms with arbitrary bases.In summary, through multiple approaches—converting the logarithmic equation to exponential form, using the change of base formula, applying logarithmic identities, and considering the properties and behavior of logarithmic functions—I arrived consistently at the result of -2 for log base 1/4 of 16. Each method reinforces the correctness of the solution, and understanding these different perspectives deepens my grasp of logarithmic concepts.One last check: if I plug x = -2 back into the original logarithmic expression, does it hold true? Let's see:log_(1/4)(16) = -2 implies that (1/4)^(-2) = 16. Calculating (1/4)^(-2), which is the same as (4^(-1))^(-2) = 4^(2) = 16. Yes, that's correct. So, the solution satisfies the original equation.This thorough exploration has helped me not only solve the problem but also understand the underlying principles that make the solution valid. It's a good reminder of how logarithms work, especially with fractional bases, and how different mathematical tools and properties can be used to verify solutions.

💡First, I need to understand the relationship between the height and volume of a cone. The volume of a cone is directly proportional to its height when the radius remains constant.Given that the volume increases by 160%, this means the new volume is 260% of the original volume.Let’s denote the original height as ( h ) and the new height as ( h' ). The ratio of the new volume to the original volume is equal to the ratio of the new height to the original height.So, ( frac{V'}{V} = frac{h'}{h} ).Substituting the given volume increase, ( frac{260%}{100%} = frac{h'}{h} ).This simplifies to ( 2.6 = frac{h'}{h} ).To find the percentage increase in height, I subtract 1 from 2.6, which gives 1.6.Converting this to a percentage, the height was increased by 160%.

💡Okay, so I have this problem here: I need to determine the fourth term in the expansion of (left( frac{1}{x} + x right)^n), and I'm given that the binomial coefficients of the third and seventh terms are equal. Hmm, let me try to figure this out step by step.First, I remember that in a binomial expansion of ((a + b)^n), the general term is given by (T_{k+1} = binom{n}{k} a^{n-k} b^k). So, in this case, (a = frac{1}{x}) and (b = x). That means each term in the expansion will look like (binom{n}{k} left( frac{1}{x} right)^{n - k} x^k).Wait, let me write that out more clearly. The general term (T_{k+1}) is:[T_{k+1} = binom{n}{k} left( frac{1}{x} right)^{n - k} x^k]Simplifying that, since (left( frac{1}{x} right)^{n - k} = x^{-(n - k)}), so multiplying by (x^k) gives:[T_{k+1} = binom{n}{k} x^{-(n - k) + k} = binom{n}{k} x^{2k - n}]Okay, so each term is (binom{n}{k} x^{2k - n}).Now, the problem says that the binomial coefficients of the third and seventh terms are equal. Let me figure out what (k) corresponds to the third and seventh terms.In the expansion, the first term is (k = 0), so the third term is (k = 2) and the seventh term is (k = 6). Therefore, the binomial coefficients for these terms are (binom{n}{2}) and (binom{n}{6}), respectively.Given that these coefficients are equal:[binom{n}{2} = binom{n}{6}]I remember that (binom{n}{k} = binom{n}{n - k}), so this equation implies that:[binom{n}{2} = binom{n}{n - 6}]But since (binom{n}{2} = binom{n}{6}), it must be that either (2 = 6) or (2 = n - 6). Well, (2 = 6) is obviously not true, so the other case must hold:[2 = n - 6 implies n = 8]Alright, so (n = 8). Now, I need to find the fourth term in the expansion of (left( frac{1}{x} + x right)^8).Using the general term formula again, the fourth term corresponds to (k = 3) (since the first term is (k = 0)):[T_4 = binom{8}{3} left( frac{1}{x} right)^{8 - 3} x^3]Simplifying the exponents:[left( frac{1}{x} right)^{5} = x^{-5}]and[x^3 = x^3]Multiplying these together:[x^{-5} times x^3 = x^{-2} = frac{1}{x^2}]Wait, hold on, that doesn't seem right. Let me check my exponent calculation again. The exponent is (2k - n), so for (k = 3):[2(3) - 8 = 6 - 8 = -2]So, the exponent is (-2), which is (x^{-2}) or (frac{1}{x^2}). Hmm, but I thought the fourth term would be positive. Maybe I made a mistake in the exponent calculation.Wait, no, let me go back to the general term:[T_{k+1} = binom{n}{k} left( frac{1}{x} right)^{n - k} x^k]So, for (k = 3):[left( frac{1}{x} right)^{8 - 3} = left( frac{1}{x} right)^5 = x^{-5}]and[x^3 = x^3]Multiplying these together:[x^{-5} times x^3 = x^{-2}]So, yes, the exponent is (-2). Therefore, the fourth term is:[binom{8}{3} x^{-2}]Calculating the binomial coefficient:[binom{8}{3} = frac{8!}{3!(8 - 3)!} = frac{8 times 7 times 6}{3 times 2 times 1} = 56]So, the fourth term is:[56 x^{-2} = frac{56}{x^2}]Wait, but the problem asks for the fourth term in the expansion. Is there a different way to express this? Or maybe I need to write it as (56x^{-2}) or (frac{56}{x^2})?Looking back at the problem statement, it just says "the fourth term in the expansion," so either form should be acceptable. However, sometimes problems prefer positive exponents, so maybe I should write it as (frac{56}{x^2}).But hold on, let me double-check my steps to make sure I didn't make any mistakes.1. I identified that the third term corresponds to (k = 2) and the seventh term to (k = 6).2. Set (binom{n}{2} = binom{n}{6}), leading to (n = 8).3. Then, for the fourth term, (k = 3), so the term is (binom{8}{3} x^{-2}).4. Calculated (binom{8}{3} = 56).Everything seems to check out. So, the fourth term is indeed (56x^{-2}) or (frac{56}{x^2}).But wait, another thought: sometimes, in binomial expansions, the terms are written in descending powers of (x). So, in (left( frac{1}{x} + x right)^8), the first term is (frac{1}{x^8}), the second term is (8x^{-6}), the third term is (28x^{-4}), the fourth term is (56x^{-2}), and so on. So, the fourth term is indeed (56x^{-2}).Alternatively, if we were to write it as a positive exponent, it would be (56x^{-2}), but since the problem doesn't specify, both forms are correct. However, in the context of binomial expansions, negative exponents are standard when dealing with terms like (frac{1}{x}).So, I think I'm confident that the fourth term is (56x^{-2}), which can also be written as (frac{56}{x^2}).But just to be thorough, let me write out the expansion up to the fourth term to make sure.The expansion of (left( frac{1}{x} + x right)^8) is:1. (k = 0): (binom{8}{0} left( frac{1}{x} right)^8 x^0 = 1 times frac{1}{x^8} times 1 = frac{1}{x^8})2. (k = 1): (binom{8}{1} left( frac{1}{x} right)^7 x^1 = 8 times frac{1}{x^7} times x = 8 times frac{1}{x^6} = frac{8}{x^6})3. (k = 2): (binom{8}{2} left( frac{1}{x} right)^6 x^2 = 28 times frac{1}{x^6} times x^2 = 28 times frac{1}{x^4} = frac{28}{x^4})4. (k = 3): (binom{8}{3} left( frac{1}{x} right)^5 x^3 = 56 times frac{1}{x^5} times x^3 = 56 times frac{1}{x^2} = frac{56}{x^2})Yes, that matches what I found earlier. So, the fourth term is indeed (frac{56}{x^2}) or (56x^{-2}).I think I've thought this through thoroughly. I don't see any mistakes in my reasoning, so I'm confident that the fourth term is (56x^{-2}).

💡Alright, so I have this statement: "For all x in the interval [-1, 1], x squared plus 3x minus 1 is greater than 0." And I need to find its negation. Hmm, okay. Let me think about how to approach this.First, I remember that the negation of a universal statement ("for all") is an existential statement ("there exists"). So, if the original statement is saying that something is true for every x in this interval, the negation would say that there's at least one x in that interval where the statement isn't true.So, the original statement is: For all x in [-1, 1], x² + 3x - 1 > 0.The negation should be: There exists an x in [-1, 1] such that x² + 3x - 1 is not greater than 0. In other words, x² + 3x - 1 ≤ 0.Okay, that makes sense. But maybe I should check if this negation is actually true or not. Let me see.To verify if there's such an x, I can solve the inequality x² + 3x - 1 ≤ 0.First, I'll write it as x² + 3x - 1 ≤ 0.This is a quadratic inequality. To solve it, I need to find the roots of the equation x² + 3x - 1 = 0.Using the quadratic formula: x = [-b ± sqrt(b² - 4ac)] / (2a).Here, a = 1, b = 3, c = -1.So, discriminant D = b² - 4ac = 9 - 4(1)(-1) = 9 + 4 = 13.Since D is positive, there are two real roots.Calculating the roots:x = [-3 + sqrt(13)] / 2 ≈ (-3 + 3.6055) / 2 ≈ 0.6055 / 2 ≈ 0.30275x = [-3 - sqrt(13)] / 2 ≈ (-3 - 3.6055) / 2 ≈ (-6.6055) / 2 ≈ -3.30275So, the roots are approximately x ≈ 0.30275 and x ≈ -3.30275.Now, since the coefficient of x² is positive (1), the parabola opens upwards. This means the quadratic expression x² + 3x - 1 is negative between its two roots.So, the solution to x² + 3x - 1 ≤ 0 is x ∈ [-3.30275, 0.30275].But our interval of interest is x ∈ [-1, 1].So, the overlap between [-3.30275, 0.30275] and [-1, 1] is [-1, 0.30275].Therefore, there exists an x in [-1, 0.30275] such that x² + 3x - 1 ≤ 0.So, the negation is: There exists an x in [-1, 0.30275] such that x² + 3x - 1 ≤ 0.Wait, but in the original interval, it's [-1, 1]. So, the exact interval where the inequality holds is from -1 up to approximately 0.30275.Therefore, the negation is: There exists an x in [-1, 0.30275] such that x² + 3x - 1 ≤ 0.I think that's it. To make it precise, maybe I should write it using exact values instead of approximations.The exact roots are x = [-3 ± sqrt(13)] / 2.So, the interval where the inequality holds is x ∈ [(-3 - sqrt(13))/2, (-3 + sqrt(13))/2].But since we're only interested in x ∈ [-1, 1], the relevant interval is x ∈ [-1, (-3 + sqrt(13))/2].Calculating (-3 + sqrt(13))/2:sqrt(13) is approximately 3.6055, so (-3 + 3.6055)/2 ≈ 0.6055/2 ≈ 0.30275.So, the exact upper bound is (-3 + sqrt(13))/2, which is approximately 0.30275.Therefore, the negation is: There exists an x in [-1, (-3 + sqrt(13))/2] such that x² + 3x - 1 ≤ 0.I think that's the most precise way to write it.Just to double-check, let's pick a value in [-1, 0.30275], say x = 0.Plugging into x² + 3x - 1: 0 + 0 - 1 = -1, which is indeed ≤ 0.Another value, x = -1:(-1)^2 + 3*(-1) - 1 = 1 - 3 - 1 = -3, which is also ≤ 0.And at x = 0.30275:(0.30275)^2 + 3*(0.30275) - 1 ≈ 0.0916 + 0.90825 - 1 ≈ 1.0 - 1.0 ≈ 0.So, it checks out.Therefore, the negation is correct.

💡Okay, so I have this geometry problem here about a parallelogram and some equilateral triangles constructed on its sides. The goal is to prove that another triangle formed by certain points is also equilateral. Let me try to break this down step by step.First, let me visualize the problem. There's a parallelogram ABCD. I know that in a parallelogram, opposite sides are equal and parallel. So, AB is parallel and equal to CD, and AD is parallel and equal to BC. Now, on sides BC and CD, equilateral triangles BCP and CDQ are constructed externally. That means these triangles are built outside the parallelogram, right?So, BCP is an equilateral triangle on BC, and CDQ is another equilateral triangle on CD. I need to prove that triangle APQ is equilateral. Hmm, okay. Let me think about how to approach this.Maybe I can use some properties of parallelograms and equilateral triangles. Since ABCD is a parallelogram, I can assign coordinates to the points to make things more concrete. Let me try that.Let me place point A at the origin (0,0). Since it's a parallelogram, I can let point B be at (a,0), point D at (0,b), and then point C would be at (a,b). That way, sides AB and DC are both length a, and sides AD and BC are both length b.Now, let's construct the equilateral triangles BCP and CDQ externally. Let me figure out the coordinates of points P and Q.Starting with triangle BCP. Since B is at (a,0) and C is at (a,b), the side BC is vertical. To construct an equilateral triangle externally on BC, point P should be either to the left or right of BC. Since it's external, I think it should be to the right, but I need to confirm.Wait, actually, in a parallelogram, BC is from (a,0) to (a,b). So, constructing an equilateral triangle externally would mean rotating BC by 60 degrees. Let me recall that rotating a point around another point can help find the coordinates of P.Similarly, for triangle CDQ, which is constructed externally on CD. CD goes from (a,b) to (0,b), so it's a horizontal line. Rotating CD by 60 degrees externally will give me point Q.Hmm, maybe using vectors or complex numbers could simplify this. Let me think about complex numbers because rotations are easier to handle there.Let me represent the points as complex numbers. Let me denote A as 0, B as a, D as bi, and C as a + bi.Now, to construct equilateral triangle BCP externally. So, point P is obtained by rotating vector BC by 60 degrees. Vector BC is C - B = (a + bi) - a = bi. Rotating bi by 60 degrees counterclockwise would be multiplying by e^{iπ/3} = cos(60°) + i sin(60°) = 0.5 + i (√3/2).So, vector BP = BC rotated by 60°, which is bi * (0.5 + i √3/2) = 0.5bi - (√3/2)b.Therefore, point P is B + BP = a + (0.5bi - (√3/2)b) = a - (√3/2)b + 0.5bi.Similarly, for triangle CDQ. Vector CD is D - C = (bi) - (a + bi) = -a. Rotating vector CD by 60 degrees externally. Since CD is from C to D, which is (-a, 0). Rotating this by 60 degrees counterclockwise would be multiplying by e^{iπ/3} again.So, vector CQ = CD rotated by 60°, which is (-a) * (0.5 + i √3/2) = -0.5a - i (√3/2)a.Therefore, point Q is C + CQ = (a + bi) + (-0.5a - i (√3/2)a) = 0.5a + bi - i (√3/2)a.Wait, let me compute that again. Point Q is C + CQ. C is a + bi, and CQ is -0.5a - i (√3/2)a. So, adding them together:Real part: a - 0.5a = 0.5aImaginary part: bi - i (√3/2)a = i (b - (√3/2)a)So, point Q is 0.5a + i (b - (√3/2)a).Similarly, point P was a - (√3/2)b + 0.5bi.Now, I need to find points A, P, and Q. Point A is 0, point P is a - (√3/2)b + 0.5bi, and point Q is 0.5a + i (b - (√3/2)a).I need to compute the distances AP, AQ, and PQ to see if they are equal, which would prove that triangle APQ is equilateral.First, let's compute AP. Since A is 0, AP is just the magnitude of P.AP = |P| = sqrt[(a - (√3/2)b)^2 + (0.5b)^2]Let me compute that:= sqrt[(a^2 - √3 ab + (3/4)b^2) + (0.25b^2)]= sqrt[a^2 - √3 ab + (3/4 + 1/4)b^2]= sqrt[a^2 - √3 ab + b^2]Okay, that's AP.Now, AQ is the magnitude of Q.AQ = |Q| = sqrt[(0.5a)^2 + (b - (√3/2)a)^2]Compute that:= sqrt[(0.25a^2) + (b^2 - √3 ab + (3/4)a^2)]= sqrt[0.25a^2 + b^2 - √3 ab + 0.75a^2]= sqrt[(0.25 + 0.75)a^2 + b^2 - √3 ab]= sqrt[a^2 + b^2 - √3 ab]Wait, that's the same as AP. So AP = AQ.Now, let's compute PQ. PQ is the distance between P and Q.P is a - (√3/2)b + 0.5bi, and Q is 0.5a + i (b - (√3/2)a).So, PQ = |Q - P|.Let me compute Q - P:Real part: 0.5a - (a - (√3/2)b) = 0.5a - a + (√3/2)b = -0.5a + (√3/2)bImaginary part: (b - (√3/2)a) - 0.5b = 0.5b - (√3/2)aSo, Q - P = (-0.5a + (√3/2)b) + i (0.5b - (√3/2)a)Now, the magnitude of this is sqrt[(-0.5a + (√3/2)b)^2 + (0.5b - (√3/2)a)^2]Let me compute each part:First term: (-0.5a + (√3/2)b)^2 = (0.25a^2 - (√3/2)ab + (3/4)b^2)Second term: (0.5b - (√3/2)a)^2 = (0.25b^2 - (√3/2)ab + (3/4)a^2)Adding them together:0.25a^2 - (√3/2)ab + (3/4)b^2 + 0.25b^2 - (√3/2)ab + (3/4)a^2Combine like terms:a^2 terms: 0.25 + 0.75 = 1a^2b^2 terms: (3/4 + 0.25) = 1b^2ab terms: - (√3/2)ab - (√3/2)ab = -√3 abSo, total is sqrt[a^2 + b^2 - √3 ab]Which is the same as AP and AQ.Therefore, AP = AQ = PQ, so triangle APQ is equilateral.Wait, that seems to work out. So, by assigning coordinates and computing the distances, I was able to show that all sides of triangle APQ are equal. Therefore, triangle APQ is equilateral.I think this approach is solid, but maybe there's a more geometric way without coordinates. Perhaps using rotations or properties of parallelograms and equilateral triangles. But since the coordinate method worked, I think that's a valid proof.Another thought: Maybe using vectors and showing that the vectors AP and AQ have the same magnitude and the angle between them is 60 degrees, which would imply an equilateral triangle. But I think the coordinate method was straightforward.So, in conclusion, by assigning coordinates to the parallelogram and computing the necessary distances, we've shown that triangle APQ has all sides equal, hence it's equilateral.