Cool geometry

Geometry Level 3

Let A B C ABC be a triangle in which A B = A C AB = AC and let I I be its incentre. Suppose B C = A B + A I BC = AB + AI . Find the value of ∠ B A C \angle BAC in degrees.


The answer is 90.

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Soumyajit Banerjee by Soumyajit Banerjee , 20, India

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2 solutions

汶汶 樂
Sep 19, 2015

1 Helpful 0 Interesting 0 Brilliant 0 Confused
Ajit Athle
May 18, 2018

Refer to the given diagram. Let BK = y. Then AK = √(x²-y²) and A I I K \frac{AI}{IK} = x y \frac{x}{y} or A I A K \frac{AI}{AK} = x ( x + y ) \frac{x}{(x+y)} . In other words, AI = x √ ( x ² − y ² ) ( x + y ) \frac{x√(x²-y²)}{(x+y)} and by the given condition, x + x √ ( x ² − y ² ) ( x + y ) \frac{x√(x²-y²)}{(x+y)} =2y. We solve this to get y= x √ 2 \frac{x}{√2} . Therefore, BC=√2 x which implies AB²+AC²=BC² or that angle BAC=90°

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