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I'm relying only on the geometry I learned in high school.

Given a scalene obtuse triangle $ABC$, where $AC$ is opposite the obtuse angle, and a point $D$ in $AC$ such that $AD = DC$ (a midpoint).

Then, construct line segment $BD$, subdividing the triangle. What I'm wondering about is whether the length of $BD$ is always, never, or only sometimes the same as the lengths $AD$? Are there any characteristics predictable about the subdivided triangles $ADB$ and $CDB$?

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  • $\begingroup$ My thanks to you, Shaun; I'm still sorting through how to do that. I would have preferred to draw a picture but don't know how others have done it. $\endgroup$ – Rob Perkins Jan 8 '14 at 19:03
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$BD$ is never the same as $AD$. In fact, if we drop the requirement of angle $B$ being obtuse, the condition $BD = AD$ is equivalent to the triangle being a right triangle at $B$.

The triangles $ADB$ and $CDB$ have the same base and height, hence the same area. Also, their angles at $D$ are supplementary.

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  • $\begingroup$ They have the same area, but are not congruent, right? $\endgroup$ – Rob Perkins Jan 8 '14 at 19:10
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    $\begingroup$ No, they wouldn't (at least if the vertices are ordered as $ADB$ and $CDB$). Congruence would be equivalent to having right angles at $D$, which would imply that $ABC$ was isosceles, contrary to your assumption. $\endgroup$ – user119908 Jan 8 '14 at 19:16

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