Triangles ADK and ADB are congruent by ASA, so KD=DB. Also, CM=MB by the definition of a median, so CD = 2 MD + KD. Let CAM=MAK=KAD=DAB=x. Then

2 tan 2x + tan x = 2 (MD/AD) + (KD/AD) = CD/AD = tan 3x.

However, the addition formula for tangent gives us

tan 2x = (tan 3x - tan x)/(1 + tan x tan 3x)

and substituting this in the last equation tells us that either tan x = tan 3x or tan x tan 3x = 1, i.e., either tan x = tan 3x or tan 3x = tan (pi/2 - x). Since x is between 0 and pi/4 exclusive, the first alternative is impossible and the second implies that 4x = pi/2, so BAC is right, as desired.

David Moews ( )

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