Geometry Problem. Post your solution in the comments box below.

Level: Mathematics Education, High School, Honors Geometry, College.

This entry contributed by Erina-NJ.

Click the figure below to see the complete problem 923.

## Friday, September 13, 2013

### Geometry Problem 923: Quadrilateral, Quadrangle, Obtuse Angle, Triangle, Diagonal, Inequality, Length

Labels:
diagonal,
inequality,
obtuse,
quadrilateral,
triangle

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http://img593.imageshack.us/img593/8753/f4i4.png

ReplyDeleteDraw circle diameter AC and circle ABC

In circle ABC draw diameter BD’.

Since ∠B >90 => B will be inside red circle and BD’>AC

Since ∠A and ∠C are obtuse => D will be on area xD’y

Note that ∠CD’y=∠AD’x=∠B => all these angles are obtuse

So angle(CD’D) and angle(AD’D) are obtuse => BD> BD’

And BD> AC

Let L, M be the feet of the perpendiculars drawn from B upon the rays DA, DC respectively.Since angle BAD is obtuse, A lies between L and D. Similarly, C lies between M and D. So both the points A and C lie completely inside the circle through B, L, D, N on BD as diameter. Thus the segment AC is completely inside the circle and hence its length is less than the diameter BD.

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