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

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

Click the figure below to view the complete problem 989

## Wednesday, March 5, 2014

### Geometry Problem 989: Arbelos, Semicircles, Diameter, Perpendicular, 90 Degree, Common External Tangent, Rectangle, Midpoint of Arc

Labels:
90,
arbelos,
common tangent,
degree,
diameter,
midpoint,
perpendicular,
rectangle,
semicircle

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http://s25.postimg.org/vsrf9vrnj/Prob_989.png

ReplyDeleteLet M, L, N, O are points shown on the sketch

1. We have MG⊥GH and NH⊥GH

Since LG=LB=LH=> L is the midpoint of GH

∠(GBH)= ½(∠GMB+∠BNH)= 90

And ∠ (MLN)=1/2(∠GLB)+ ∠BLH)=90

In right triangle MLN , BL^2=BM.BN

In right triangle ADC , DB^2=BA.BC= 4. BM.BN=> L is the midpoint of BD

And DGBH is a parallelogram with right angle ∠GBH => DGBH is a rectangle

2. Connect AG and CH

We have ∠ (AGB=∠ (BGD)= 90 => A,G,D are collinear

Triangles AMG and AOD are isosceles with common angle ∠GAM

So ∠ (AGM)= ∠ (ADO) => OD//MG => OD ⊥ EF

So D is the midpoint of arc EF

I have a comment on the second part.

ReplyDeleteDO || HN, and they are perpendicular to EF.

So DO bisects both chord EF and arc EF.

Thank you for your comment on my solution.

DeletePeter

Because <GAC=<FHC, AGHC is cyclic. Let AG and HC meet at D'. <D'AC+<D'CA=90, so D' must be on big circle. Furthermore, because D'G*D'A=D'H*D'C from cyclic property of AGHC, D' must also be on radical axis of 2 small circles, which is their common tangent. Therefore, D' coincides with D. It is easy to see that GDHB has 3 right angles from Thales theorem in each circle and must therefore be rectangle.

ReplyDelete<GAC=<FHC=arcDF/2+arcFC/2=arcDE+arcFC/2.

This means that arcDF=arcDE so D is midpoint of arcEF.

Draw a circle with diameter BD. It meets (AB) in X and (BC) in Y.

ReplyDelete<BXD=<BYD=90 deg by Thales' theorem.

<BXA=<BYC=90 deg by Thales' theorem.

So X lies on AD and Y lies on BD.

Thales again: <XDY = <ADC = 90 deg.

Now note that <BXY = 90 deg - <BDX = <BAX.

So XY is tangent to the (AB) at X and similarly tangent to (BC) at Y. So X=G and Y=H.

By similarity the tangent to the (AC) at D is parallel to GH. Let M be the midpoint of {AC), then MD is perpendicular to EF, hence bisects EF and bisects angle EMF, and D is midpoint of arc EF.

Let GH, BD cut at X. Let O be the centre of AC and let OD cut GH at Y.

ReplyDeleteLet AB = 2a, BC = 2b and let GH = 2t,

Now XG = XB = XH = t. Also GF^2 = (a-b)^2 + (a+b)^2 = 4ab = 4t^2 = BD^2 hence BX = FX = DX = GX = t. It follows that BFDG is a rectangle

If < DAO = p, < DOB = p = < BXH

Therefore OYXB is concyclic and so OY is perpendicular to EG

It follows that EY = YF and hence ED = DF which is the result we need

Sumith Peiris

Moratuwa

Sri Lanka

https://photos.app.goo.gl/Ps4iad2ZZNznvwP17

ReplyDelete