Angle Bisectors In Rectangle

What is this about?
A Mathematical Droodle


This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at https://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.


What if applet does not run?

Explanation

|Activities| |Contact| |Front page| |Contents| |Geometry|

Copyright © 1996-2018 Alexander Bogomolny

The applet attempts to suggest the following problem [Prasolov, p. 12]:

ABCD is a rectangle; M and N are the midpoints of sides AD and BC, respectively. Let P lie on CD, Q be the intersection of MP and AC. Prove that MN is the bisector of ∠PNQ.


This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at https://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.


What if applet does not run?

Since MN||CD triangles CPQ and OMQ are similar, implying the proportion

MP/MQ = CO/OQ.

Let O be the midpoint of AC (i.e., the center of the rectangle) and K on NQ be such that KO||BC. Then triangles CNQ and OKQ are similar, implying the proportion

CO/OQ = KN/KQ.

By the transitivity of equality, we have a proportion in triangles NPQ and KMQ

MP/MQ = KN/NQ.

which implies KM||NP. We are almost done.

ΔKMN is isosceles (because MO = NO and KO⊥MN) so that ∠KMN = ∠KNM. But, since KM||NP, the vertical angles KMN and MNP are also equal, giving the required ∠QNM = ∠MNP.

References

  1. V. V. Prasolov, Problems in Planimetry I, Nauka, 1986 (in Russian)

Related material
Read more...

  • Segment Trisection Induced by Parallels to Medians
  • The Nature of Pi
  • Parallelogram and Similar Triangles
  • Two Triples of Similar Triangles
  • Three Similar Triangles
  • Similar Triangles on Sides and Diagonals of a Quadrilateral
  • Point on Bisector in Right Angle
  • |Activities| |Contact| |Front page| |Contents| |Geometry|

    Copyright © 1996-2018 Alexander Bogomolny

     63039598

    Search by google: