Euler is Everywhere! #3
Triangle with area is inscribed in a circle with radius A point is selected such that it is units away from the center of this circle. Let be the feet of the perpendiculars dropped from to the sides of
Find the area of
The answer is 4.
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1 solution
We call A ′ B ′ C ′ the pedal triangle with respect to P . Let R be the length of the radius of the circle and O be its center. A theorem from Euler states that the area of A ′ B ′ C ′ equals
∣ ∣ ∣ ∣ 4 R 2 R 2 − O P 2 ∣ ∣ ∣ ∣
times the area of A B C . Since R = 5 , O P = 3 , and [ A B C ] = 2 5 ,
[ A ′ B ′ C ′ ] = ∣ ∣ ∣ ∣ 4 ( 5 2 ) 5 2 − 3 2 ∣ ∣ ∣ ∣ ( 2 5 ) = 2 5 4 ( 2 5 ) = 4 .