1. ## circle (2)

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 .

2. Hello thereddevils
Originally Posted by thereddevils
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 $\displaystyle QS$ (which I think is what you meant) looks more like the bisector of $\displaystyle \angle PSR$.

Then in $\displaystyle \triangle$'s $\displaystyle SQP, SQR$
$\displaystyle SP = SR$ (tangents from a point to a circle)

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

$\displaystyle QS$ is common
$\displaystyle \therefore \triangle$'s $\displaystyle SQP, SQR$ are congruent (SAS)

$\displaystyle \therefore PQ=RQ$

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

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

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

Then in $\displaystyle \triangle$'s $\displaystyle SQP, SQR$
$\displaystyle SP = SR$ (tangents from a point to a circle)

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

$\displaystyle QS$ is common
$\displaystyle \therefore \triangle$'s $\displaystyle SQP, SQR$ are congruent (SAS)

$\displaystyle \therefore PQ=RQ$

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

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

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

4. Hello thereddevils
Originally Posted by thereddevils
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.

5. ## circle (2)

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