Problem 3 : The Mysterious Triangle

Geometry Level 4

Let $ABC$ be a triangle. Let $D$ be on ray $BC$ such that $BC = CD$ . Let $E$ be on ray $CA$ such that $AE = 2CA$ .Let $AD = BE$ . Determine the largest angle of $\triangle ABC$ .

The answer is 90.

1 solution

Zi Song Yeoh
Jul 28, 2014

It is easy to see that $A$ is the centroid of triangle $BDE$ . The midpoint of $BE$ is thus the circumcenter of triangle $ABE$ , which implies $ABC$ is right-angled. Thus, the answer is $\boxed{90}$ .

Cool, I used a bunch of trigonometry for nothing.. $\begin{cases} \text{AD}^2 = a^2 + b^2 + 2ab \cos \gamma \\ \text{BE}^2 = 4b^2 + c^2 + 4bc \cos \alpha\\ c^2 = a^2 + b^2 - 2 ab \cos \gamma \\ a^2 = b^2 + c^2 - 2bc \cos \alpha \\ \text{AD}^2 = \text{BE}^2 \end{cases}$ from that system it follows that $\cos \alpha = 0$ .

Paolo Bentivenga - 6 years, 10 months ago

Problem 3 : The Mysterious Triangle

The answer is 90.

1 solution

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