Use the twocolumn format A circle inscribed in Delta ABC is

Use the two-column format A circle inscribed in Delta ABC is tangent to sides BC, CA and AB at points L, M, and N, respectively Line MN intersects line BC at P, line NL intersects line AC at Q, and line ML intersects line AB at R. Prove that P, Q, and R are collinear.

Solution

Suppose, P,Q,R are not colinear and P,Q and R\' are colinear.

so, triangle PLR\' have three sides PL,LR\', R\'P.

PR\' line is through Q.

so, PR\' = PQ + QR\'

that means, LM intersect AB at R\' and PQ intersect LM at R\'.

but given that LM intersect AB at R.

Now, two straight line only intersect once, then can never intersect twice.

so, after intersecting at R that will not intersect in R\'.

so, LM intersect PQ at R.

Here, P, Q, R are at same line (or colinear)....... PROVED