MathLabs

Problem 1

In triangle ABCABC, let D,E,FD,E,F be the midpoints of BC,CA,ABBC,CA,AB, respectively, and let GG be the centroid. For each value of ∠BAC\angle BAC, how many non-similar triangles are there for which AEGFAEGF is a cyclic quadrilateral?
Step 1 of 5: Record the midpoint and centroid ratios
Previous stepNext step
AI:AD=2:3,IG:AD=1:3,FI:IE=1:4AI:AD=2:3,\quad IG:AD=1:3,\quad FI:IE=1:4
Detailed analysis

Let I=AG∩EFI=AG\cap EF. Since EF∥BCEF\parallel BC and GG is the centroid, the standard midpoint and centroid ratios give AI:AD=2:3AI:AD=2:3, IG:AD=1:3IG:AD=1:3, and FI:IE=1:4FI:IE=1:4.