Category Archives: Trigonometry

AIME 2017 I – Problem 15

Posted on December 31, 2022 by kevin

The area of the smallest equilateral triangle with one vertex on each of the sides of the right triangle with side lengths $2\sqrt{3}$, $5$, and $\sqrt{37}$ as shown, is $\dfrac{m\sqrt{p}}{n}$, where $m$, $n$, and $p$ are positive integers, and $m$, $n$, … Continue reading →

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Distance between the incenter and circumcenter

Posted on October 7, 2022 by kevin

The distance between the incenter and circumcenter of a triangle, is calculated by Euler’s theorem in geometry: $$d=\sqrt{R(R-2r)}$$ which also implies $$R\ge2r$$ For a Bicentric quadrilateral, the distance between the incenter and circumcenter can be calculated by Fuss’s theorem or … Continue reading →

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Trigonometry Challenge 2022/09/26

Posted on September 26, 2022 by kevin

Let $\alpha$, $\beta$, and $\gamma$ be the interior angles of $\triangle{ABC}$. Find all solutions so that $$\cos\alpha\cdot\cos\beta+\sin\alpha\cdot\sin\beta\cdot\sin\gamma=1$$ 🔑 Solution: Since $\alpha$, $\beta$, and $\gamma$ are the interior angles of $\triangle{ABC}$, we have $$0\lt\alpha,\ \beta,\ \gamma\lt\pi$$ Therefore $$0\lt\sin\alpha,\ \sin\beta,\ \sin\gamma\le 1$$ … Continue reading →

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Probability Challenge 2022/08/29

Posted on August 29, 2022 by kevin

A frog can randomly jump exactly 1 yard away in any random direction constantly. (1) What is the probability that the frog is within 1 yard away from its starting point after 2 jumps? (2) What is the probability that … Continue reading →

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Geometry Challenge – 9

Posted on February 21, 2022 by kevin

In the diagram below, two tangent semi-circles centered at A and B are inscribed in a large semi-circle center at O, and circle C is tangent to semi-circle A, B and O. Point A, O, and B are on the … Continue reading →