See complete Problem 145

Four Triangles, Incircle, Tangent and Parallel to Side, Incenters, Circumcenters. Level: High School, SAT Prep, College geometry

Post your solutions or ideas in the comments.

## Friday, July 25, 2008

### Elearn Geometry Problem 145

Labels:
circumcenter,
incenter,
parallel,
similarity,
tangent,
triangle

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1.

ReplyDeleteAll 4 triangles are similar with all corresponding sides parallel to each others', they are all homothetic to each others. As unique triangle has unique incenter and circumcenter, all shapes of triangles with inscribed and outscribed circles are homothetic to each others, thus d//d1//d2//d3 is obviously true.

2.

Let W, X, Y and Z be points on DE, HG, EF and MH respectively touching ABC's inscribed circle.

As WIX and YIZ are collinear, angle bisector EIH is a straight line, so are DIG and MIF.

As IDE and IGH are two similar triangles with same corresponding heights, they are congruent and DE=HG.

d:d1:d2:d3=AC:AH:HG:GC => d=d1+d2+d3