Problems Involving Triangles
Posted: Wed Nov 02, 2011 12:23 pm
1. Denote by P the perimeter of triangle ABC. If M is a point in the interior of the triangle, prove that:
\[\frac{1}{2}P < MA + MB + MC < P\]
2. The side length of the equilateral triangle ABC equals l. The point P lies in the interior of ABC and the distances from P to the triangle’s sides are 1, 2, 3. Find the possible values of l.
3. Let P be a point in the interior of the triangle ABC. The reflections of P across the midpoints of the sides BC, CA, AB, are\[P_{a}, P_{b}, P_{c}\] respectively. Prove that the lines APA, BPB, and CPC are concurrent.
\[\frac{1}{2}P < MA + MB + MC < P\]
2. The side length of the equilateral triangle ABC equals l. The point P lies in the interior of ABC and the distances from P to the triangle’s sides are 1, 2, 3. Find the possible values of l.
3. Let P be a point in the interior of the triangle ABC. The reflections of P across the midpoints of the sides BC, CA, AB, are\[P_{a}, P_{b}, P_{c}\] respectively. Prove that the lines APA, BPB, and CPC are concurrent.