Problem 4

Let

ABCD

be a quadrilateral with an incircle

\omega

of centre

I

. The diagonals

AC

and

BD

intersect at

E

. Let

J

be the incentre of triangle

ABD

. The extension of the ray

EJ

intersects

\omega

P

. Prove that

PI\perp BD

Step 4 of 6: A harmonic range from the pole–polar configuration

Previous step Next step

V:=IL\cap BD \implies (P',V,Q,L)\text{ harmonic}

Detailed analysis

Let $V:=IL\cap BD$ . Since $BD$ is the polar of $L$ with respect to $\omega$ , and $P'Q$ is the diameter of $\omega$ perpendicular to $BD$ , the standard harmonic property of poles and polars gives that $P',V,Q,L$ form a harmonic range on line $IL$ . Because inversion and projection from a point both preserve cross-ratio, this harmonic relation transfers to the pencils through $J$ and through $A$ .