A quarter circle

Geometry Level 1

O A B OAB is a quarter circle, and points H 1 H_1 and H 2 H_2 divide the arc into three equal parts. Now, we drop perpendiculars from H 1 H_1 and H 2 H_2 to O A OA .

Which is larger, the blue triangle or orange quadrilateral?

Blue triangle Orange quadrilateral They are equal

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Áron Bán-Szabó by Áron Bán-Szabó , 17, Hungary

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2 solutions

This is a proof without words solution:

7 Helpful 0 Interesting 0 Brilliant 0 Confused
Marta Reece
Jun 8, 2017

T = [ O H 1 G 1 ] − [ O K G 2 ] T=[OH_1G_1]-[OKG_2]

t = [ O H 2 G 2 ] − [ O K G 2 ] t=[OH_2G_2]-[OKG_2]

T − t = [ O H 1 G 1 ] − [ O H 2 G 2 ] T-t=[OH_1G_1]- [OH_2G_2]

[ O H 1 G 1 ] = 1 2 sin ⁡ 3 0 ∘ cos ⁡ 3 0 ∘ O B ‾ 2 = 3 8 O B ‾ 2 [OH_1G_1]= \frac12\sin30^\circ\cos30^\circ\overline{OB}^2=\dfrac{\sqrt{3}}{8}\overline{OB}^2

[ O H 2 G 2 ] = 1 2 sin ⁡ 6 0 ∘ cos ⁡ 6 0 ∘ O B ‾ 2 = 3 8 O B ‾ 2 [OH_2G_2]= \frac12\sin60^\circ\cos60^\circ\overline{OB}^2=\dfrac{\sqrt{3}}{8}\overline{OB}^2

T − t = [ O H 1 G 1 ] − [ O H 2 G 2 ] = 0 T-t=[OH_1G_1]- [OH_2G_2]=\boxed{0}

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