## Thursday, June 2, 2016

### Geometry Problem 1221: Intersecting Circles, Chord, Tangent, Parallel Chords, Collinear Points

Geometry Problem. Post your solution in the comment box below.
Level: Mathematics Education, High School, Honors Geometry, College.

Click the figure below to view more details of problem 1221.

1. Since EF//CD => Arc(EC)=Arc(FD)
In circle O , We have ∠ (DTB)= ½( Arc(EC)+Arc(DB))= ½(Arc(FD)+Arc(DB))= ½ Arc(FB)= ∠ (BAF)… (1)
Since DT tangent to circle Q at T => ∠ (BAT)= ∠ (DTB)…. (2)
Compare ( 1) to (2) we have ∠ (BAF)= ∠ (BAT)
So F, T , A are collinear

2. < DTB = < FET but < DTB = < TAB

So < FET = < TAB
But < FET = < FAB
Hence < TAB = < FAB

So, F, T, A are collinear

Sumith Peiris
Moratuwa
Sri Lanka

3. Is arc EC=arc FD. Suppose that AT intersects the circle with center O to the point Κ,then <EKA=<EBA=<ATC=<KTD so EK//CD.Therefore arc KD=arc FD, so the points K, F will coincide.
Therefore points F,T,A are collinear.

4. Consider the line FA intersects the circle Q at P. Join AB.
P and T coincide and hence F,T,A are collinear

5. Ang EFA= Ang EBA= Ang CTA, because EF//CD => F, T, A collinear