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

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

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## Sunday, March 8, 2015

### Geometry Problem 1094: Tangent Circles, Tangent Chord, Radius, Sagitta of Arc

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Observe that CDT are collinear,

ReplyDeletethus ab=CD×DT.

Now since

CD=c/cos∠MDT

DT=2r cos∠QDT

Hence, ab=2CR.

A typo I think

ReplyDeleteIt should be CD=c/cos∠MCD

(ab = 2cr of course)

Thanks Pravin. It's MCD, and ab=2cr (lower case).

DeleteGeometry solution

ReplyDeleteO,Q,T are collinear

OQ^2 = MD^2 + (QD-OM)^2

So { (a-b)/2}^2 + (r-R+c)^2 = (R-r)^2.......(1)

Applying Pythagoras to Tr. AOM

(R-c)^2 + {(a+b)/2}^2 = R^2...,,,(2)

(2) -(1) and simplifying using the difference of 2 squares

ab + r(2R-2c-r) = 2rR -r^2

Which further simplifies to

ab = 2cr

Sumith Peiris

Moratuwa

Sri Lanka