It's 7-8-9 this time

Geometry Level 4

The sides of triangle A B C ABC have lengths 7 , 8 7, 8 and 9. 9. P P is an interior point such that: ∠ P A B = ∠ P B C = ∠ P C A = θ \angle PAB=\angle PBC=\angle PCA=\theta Find the value of tan ⁡ θ \tan \theta .


The answer is 0.553.

Sathvik Acharya by Sathvik Acharya , 17, India

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1 solution

Jon Haussmann
Feb 27, 2017

Angle θ \theta is the Brocard Angle of triangle A B C ABC . The area of the triangle is K = 12 5 K = 12 \sqrt{5} , so tan ⁡ θ = 4 K a 2 + b 2 + c 2 = 4 ⋅ 12 5 7 2 + 8 2 + 9 2 = 24 5 97 . \tan \theta = \frac{4K}{a^2 + b^2 + c^2} = \frac{4 \cdot 12 \sqrt{5}}{7^2 + 8^2 + 9^2} = \frac{24 \sqrt{5}}{97}.

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