The figure below represents arbitrary triangle ABC. The points K,L,M are the midpoints of its sides.
a) Show that triangle CLK ~ triangle CAB (and similarly for the other two corner triangles) How do I go about writing a proof for this.
The definition for similarity: two figures are similar if one is congruent to the dilation of the other.
Definition of congruent: Two figures are congruent if one can be obtained from the other by a rigid motion.
b). Show that in fact all four smaller triangles (the three corner ones and the middle one) are congruent to each other by showing that they are all similar to each other with magnification factor equal to one.
Any tips on how to go about proving this? Any help is appreciated. Thanks.
I would start by showing that the large triangle ACB is similar to small triangle LCK. That they have angle C in common is a gimme.
That angle CLK = angle LAM follows from showing that segments LK and AM are parallel. That follows from the fact that CL and CK are cut off in the same proportion from CA and CB.
Now you can show that angles CKL and CKM are equal.
Having established the congruence of those 3 angles, you've demonstrated the similarity of the large and upper small triangles. You are on your way to solving the whole problem.