Geometry Problem. Post your solution in the comments box below.

Level: Mathematics Education, High School, Honors Geometry, College.

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## Saturday, November 8, 2014

### Geometry Problem 1058: Triangle, Sum of the Areas, Altitude, Perpendicular, Circumcircle, Circle, Diameter

Labels:
altitude,
area,
circle,
circumcircle,
diameter,
perpendicular,
triangle

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One can show that BD⊥EF.

ReplyDeleteThen

S(BEDF)

= 1/2×BD×EF

= BO×EF

= S(ABC) (by Problem 1055)

Thus sobtracting S(BEGMF) both side,

S₁+S₂=S₃

Join DA, DC,DH. Let (XYZ) stand for area of ΔXYZ.

ReplyDeleteAD // EH implies (AED) = (AHD). Follows S1 = (GHD).

DC // HF implies (DFC) = (DHC). Follows S2 = (MHD).

Hence S1 + S2 = S3.