circle (2)

thereddevils · #1 $Old$ November 18th, 2009, 06:42 AM

In the figure , ST is the common tangent at P to the two circles . PVW and PQR are two straight lines . OS bisects \angle PSR and PV=PS , the lines QV and RW are also parallel .

Prove that SR=VW .

Grandad · #2 $Old$ November 18th, 2009, 08:53 AM

Hello thereddevils

Quote:

Originally Posted by thereddevils $View Post$

In the figure , ST is the common tangent at P to the two circles . PVW and PQR are two straight lines . OS bisects \angle PSR and PV=PS , the lines QV and RW are also parallel .

Prove that SR=VW .

I've re-drawn the diagram (attached), so that $QS$ (which I think is what you meant) looks more like the bisector of $\angle PSR$ .

Then in $\triangle$ 's $SQP, SQR$

$SP = SR$ (tangents from a point to a circle)

$\angle PSQ = \angle SRQ$ ( $SQ$ is angle bisector, given)

$QS$ is common

$\therefore \triangle$ 's $SQP, SQR$ are congruent (SAS)

$\therefore PQ=RQ$

Now - and I'll leave you to fill in the details - prove that $\triangle$ 's $PVQ, PWR$ are similar. Hence $PV:VW=PQ:QR$ , and hence $PV=VW$ .

But $PV = PS$ (given). Therefore ... ?

Grandad

thereddevils · #3 $Old$ November 19th, 2009, 02:41 AM

Quote:

Originally Posted by Grandad $View Post$

Hello thereddevilsI've re-drawn the diagram (attached), so that $QS$ (which I think is what you meant) looks more like the bisector of $\angle PSR$ .

Then in $\triangle$ 's $SQP, SQR$

$SP = SR$ (tangents from a point to a circle)

$\angle PSQ = \angle SRQ$ ( $SQ$ is angle bisector, given)

$QS$ is common

$\therefore \triangle$ 's $SQP, SQR$ are congruent (SAS)

$\therefore PQ=RQ$

Now - and I'll leave you to fill in the details - prove that $\triangle$ 's $PVQ, PWR$ are similar. Hence $PV:VW=PQ:QR$ , and hence $PV=VW$ .

But $PV = PS$ (given). Therefore ... ?

Grandad

Thanks again Grandad , what software do you use to produce such a neat diagram ?

Grandad · #4 $Old$ November 19th, 2009, 09:14 AM

Hello thereddevils

Quote:

Originally Posted by thereddevils $View Post$

Thanks again Grandad , what software do you use to produce such a neat diagram ?

I have CorelDraw version 12, and Corel Photo-paint 12 on my maching, but they're much too heavyweight for this. So I just use the drawing tools in Word, which are very quick and easy to use. Any text I put in a text box, with no border or background colour. Then, with the image on screen, I use a screen-capture program (MWSnap - freeware, donations appreciated; works fine in Vista) to capture the image to clipboard; paste into Paint; save and then crop in Windows Photo Gallery. Sounds complicated, but it's actually very quick. Graphs I tend to draw in Excel, copy, paste, crop and upload.

Grandad

bjhopper · #5 $Old$ November 20th, 2009, 02:10 PM

originally posted by thereddevils

Hello reddevil and grandpa'

The diagrames so far do not appear correct. The initial data are not adequate

Try this
O is the center of the larger circle.The smaller circle passes thru 0
When SP=PV the diagram is unique (only one)Other diagrams SP is not equal PV but always PV=VW andPQ=QR

bjh