The Math Forum

Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Math Forum » Discussions » sci.math.* » sci.math

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

Topic: Without using the cosine rule or steps from its derivation...
Replies: 15   Last Post: May 4, 2014 10:41 AM

Advanced Search

Back to Topic List Back to Topic List Jump to Tree View Jump to Tree View   Messages: [ Previous | Next ]

Posts: 12,067
Registered: 7/15/05
Re: Without using the cosine rule or steps from its derivation...
Posted: May 2, 2014 10:39 PM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

snmp wrote:
>What is the solution to this problem as stated by the OP?

I'll restate the problem, modified slightly, as follows:


If convex cyclic pentagon PABCD is such that angles
APB,BPC,CPD are equal, then (PB + PD)/(PA + PC) = PC/PB.

Proof (ignoring restrictions on allowable methods):

Inscribe pentagon PABCD in a circle.

Since angles APB,BPC,CPD are equal, chords AB,BC,CD have
equal length, equal to v, say. Thus

AB = BC = CD = v

Since angles APC and BPD are equal, chords AC,BD have
equal length, equal to w, say. Thus

AC = BD = w

Quadrilaterals PBCD and PABC are convex cyclic, hence Ptolemy's
Theorem yields

v*PB + v*PD = w*PC

v*PA + v*PC = w*PB

Dividing the equations, we get

(PB + PD)/(PA + PC) = PC/PB

as was to be shown.


Point your RSS reader here for a feed of the latest messages in this topic.

[Privacy Policy] [Terms of Use]

© The Math Forum at NCTM 1994-2018. All Rights Reserved.