Geometry Problem

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

Click the figure below to see the problem 753 details.

## Thursday, May 17, 2012

### Problem 753: Triangle, Altitudes, Orthocenter, Sum of ratios

Labels:
altitude,
orthocenter,
ratio,
triangle

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DH/AD = (BHD)/(BAD) = (CHD)/(CAD)

ReplyDelete= Sum of numerators/sum of denominators

=[(BHD)+(CHD)]/[(BAD)+(CAD)]

=(BHC)/(ABC)

Similarly

EH/BE = (AHC)/(ABC)& FH/CF = (ABH)/(ABC)

Hence

DH/AD + EH/BE + FH/CF

=[(BHC)+(AHC)+(ABH)]/(ABC)

=(ABC)/(ABC)

= 1

Remark: Result holds for any interior point P,

not necessarily H.