Midpoints Forced by Area Ratios

★ 7/10plane geometry Menelaus's Theorem Triangle Geometry

Problem

As shown in the figure, in the quadrilateral $ABCD$ the areas of the triangles $ABD$ , $BCD$ , and $ABC$ are in the ratio

[ABD]:[BCD]:[ABC]=3:4:1.

Points $M$ and $N$ lie on the segments $AC$ and $CD$ respectively, satisfying $\dfrac{AM}{AC}=\dfrac{CN}{CD}$ , and the points $B$ , $M$ , $N$ are collinear.

Prove that $M$ and $N$ are the midpoints of $AC$ and $CD$ respectively.

Answer

Solution

Difficulty	7/10
Topics	plane geometry, Menelaus's Theorem, Triangle Geometry

Whiteboard

Your sketch is saved only in this browser. To share it, export your drawing as an image (whiteboard menu → Export as → PNG), then upload that image in the comments below.

Discussion

Ask questions, share alternate solutions, and use LaTeX freely.

0 comments

No comments yet.