What Austin Powers Can Teach You about Similar Triangles

Determining the Area of Similar Triangles on the GMAT

Looking for Similar Triangles on the GMAT
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Hi VeritasKarishma Bunuel

Had a question on this post, specifically on this bit



A few things this implies

- Should an angle bisector in a triangle which is also a median be perpendicular to the opposite side? Yes.

- Can we have an angle bisector which is also a median which is not perpendicular? No. Angle bisector which is also a median implies isosceles triangle which implies it is also the altitude.

- Can we have a median from vertex A which is perpendicular to BC but does not bisect the angle A? No. A median which is an altitude implies the triangle is isosceles which implies it is also the angle bisector.



Clarification on the purple :

Is this applicable only in the following cases

-- equilateral triangles

-- isosceles triangle where triangle ABC is isosceles such that AB = AC and BC is the base ( if the line is drawn instead from point C, instead of A drawn towards AB instead -- then the purple does not count)


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No, for all triangles.
If an angle bisector (of any triangle) is a median too, then the triangle is isosceles such that the two sides that form this angle are equal. So that angle bisector is the altitude too.
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Hi VeritasKarishma - thank you for replying.

When you say "ALL triangles" per the yellow, - i am confused. I think you are referring to a smaller subset of triangles only (i.e. isosceles / equilateral triangle only) because a scalene triangle (obtuse or acute) WILL never have the median = angle bisector to begin with.

This is a summary of what i have picked up from the initial post above.

Please let me know your thoughts !

(1) : In equilateral triangles: all medians & angle bisectors and altitudes are the same from any of the 3 angles.
(2) : In isosceles triangles : angle bisector | median | altitude of vertical angle only (non base angles) are the same. That is not true from the two non base angles.
(3) : In scalene triangles (obtuse or acute) : medians /bisectors / altitudes are ALWAYS different from each of the three angles.

If you know that in a triangle (you do not know what kind of a triangle it is), the angle bisector is the median too, then you can say that the triangle MUST BE an isosceles triangle.

The relation holds both ways:
In an isosceles triangle, the angle bisector is the median too.
In any triangle, if the angle bisector is the median too, then the triangle has to be isosceles.

The point is not that in a scalene triangle, the angle bisector can be the median - no, it cannot be. The point is that in any triangle, knowing that the angle bisector is the median tells us that the triangle has to be isosceles (could be equilateral too).
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