Question 86
Let
- $P$ be a point inside a triangle $\triangle ABC$,
- $\triangle DEF$ be the pedal triangle of $P$, i.e., let $D, E, F$ be the feet of the altitudes from $P$ to $BC, CA, AB$, respectively,
- $I$ be the incenter of $\triangle ABC$, and
- $\triangle XYZ$ be the Cevian triangle of $I$, i.e., $X, Y, Z$ be the intersections of $AI, BI, CI$ with $BC, CA, AB$, respectively.
Show that there is a triangle with side lengths $PD, PE$, and $PF$ if and only if $P$ is inside $\triangle XYZ$.
- $P$ be a point inside a triangle $\triangle ABC$,
- $\triangle DEF$ be the pedal triangle of $P$, i.e., let $D, E, F$ be the feet of the altitudes from $P$ to $BC, CA, AB$, respectively,
- $I$ be the incenter of $\triangle ABC$, and
- $\triangle XYZ$ be the Cevian triangle of $I$, i.e., $X, Y, Z$ be the intersections of $AI, BI, CI$ with $BC, CA, AB$, respectively.
Show that there is a triangle with side lengths $PD, PE$, and $PF$ if and only if $P$ is inside $\triangle XYZ$.