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Geometry ProblemLevel: High School, SAT Prep, College geometry.Click the figure below to see the complete problem 523 about Tangent Circles, Diameter Perpendicular, Collinearity.
BGD = ADG ( alternate ang )BGC = BCG, ADC = ACD =>ACD = BCG it happen when "they" are verticales angles or D, C, G are collinear
Note that points A, C and B are collinear ( Circle C1 tangent to C2 at C)BG perpendicular to GF ( FG tangent to C2 at G)So DF // BG and angle DAC=angle CBG ( alternate angles)And angle ACD= angle BCG ( Isosceles triangles DAC and CBG are similar) So Points D, C G are collinearPeter Tran
(Proof of Contradiction) Connect CE and draw common tangent at E (intersect the tangent at EF and call the intersection H). Suppose if DCG are noncollinear, then there's Quadrilateral CDFG. Angle DCE is right angle (inscribed angle on diameter), Angle ECH=Angle CDE (tangent angle equal to inscribed angle corresponding to its intercepted arc), Angle GCH=Angle CGF (transveral connecting two tangent points have equal consecutive interior angles). Add up the angles in Quadrilateral CDFG, Angles (ECH+DCE+CDE+GCH+CGF+DFG)=360 Degrees.But since DFG (given) and DCE are both right angles, Angle ECH=Angle CDE, and Angle GCH=Angle CGF, this time Angles 2(DCE+ECH+GCH)=2 Angle DCG=360 Degrees, so Angle DCG=180 Degrees, supposing a quadrilateral--contradiction! Therefore, DCG are collinear.
Let the tangents at C and G to circle B meet at H. Now HC is perpendicular to ACB. Hence HCBG & ACFH are both cyclic. So < DCA = 1/2 < CAE = 1/2 < CHF = supplementary of B / 2 = < BCG. Hence DBG are collinearSumith PeirisMoratuwaSri Lanka