Prove that the perpendiculars from $D,E,F$ to $BC,CA,AB$ are concurrent

Question

If two triangles $ABC$ and $DEF$ are such that the perpendicular from $A,B,C$ to $EF,FD,DE$ are congruent, prove that the perpendiculars from $D,E,F$ to $BC,CA,AB$ are concurrent.

Source: Challenge and Thrills of pre-college mathematics.

Thanks in advance!

score 2 · Answer 1 · answered Sep 27 '15 at 12:29

2

Hint: the perpendicular to $AB$ through $C\not\in AB$ is the locus of points $P$ for which: $$ PA^2-PB^2 = CA^2-CB^2.$$

answered Sep 27 '15 at 12:29

Jack D'Aurizio

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D' Aurizio can you help me with a diagram and explain on the technique or idea used whence I can understand and solve similar questions. – gaufler Sep 27 '15 at 12:35

Prove that the perpendiculars from $D,E,F$ to $BC,CA,AB$ are concurrent

1 Answers1