perplexus.info :: Geometry : A Cute Triangle

A Cute Triangle (Posted on 2006-07-21)

Given any triangle, through each vertex draw the external angle bisector at that vertex. Show that the new triangle that has as its vertices the three pairwise intersection points of these is always an acute triangle (all three angles strictly less than 90 degrees).

Extra Credit: Extended to meet the new triangle, the internal angle bisectors of the given triangle are what with respect to the new triangle?

Submitted by Richard

Rating: 3.5000 (2 votes)

Solution: (Hide)

Call the given triangle ABC. The external angle bisector at a vertex of ABC is perpendicular to the (ordinary) internal angle bisector at that vertex. By simple arithmetic, the internal angles at the vertices of the constructed triangle are found to be (A + B)/2, (A + C)/2, and (B + C)/2. But (A + B)/2 < 90 since A + B < 180, and similarly for the others.
It is well-known that the angle bisectors of the given triangle coincide with the altitudes of the new triangle. The given triangle is the orthic triangle of the new triangle. For an interesting slant on this see http://paideiaschool.org/TeacherPages/Steve_Sigur/resources/trilinear%20lines.pdf

Comments: ( You must be logged in to post comments.)

Subject	Author	Date
re(2): Solution	Charlie	2006-07-21 15:25:55
re: Solution	Richard	2006-07-21 14:38:53
Solution	JLo	2006-07-21 09:49:43
Solution	Bractals	2006-07-21 09:47:37

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