Revision 4

Revision Sheet 4 I.

Given a circle ( C) with center O and radius R. [OA] and [OB]are two perpendicular radii of the circle. Let C be a point on the major arc AB. The tangents at A and C to the circle intersect at point M. The lines (CM) and (OB) intersect at point P. Let D be the orthogonal projection of M on (OB). 1) Complete the figure. 2) Show that the quadrilateral MDOA is a rectangle. 3) Prove that the triangles PCO and PDM are congruent, and write their homologous parts. 4) Deduce that the triangle PMO is isosceles at P. 5) Prove that the points O, D, C, and M belong to the same circle whose center and radius are to be determined. 6) Let I be the point of intersection of (OC) and (DM). Show that (PI) is perpendicular to (OM). 7) Show that OD=CM. 8) Suppose in this part that . ˆ = 60o AOM

i) ii)

I.

Show that the triangle PCO is semiequilateral. Calculate in this case PC and PO in terms of R.

( C ) is a semi-circle of center O and diameter [AN]. E is a point of the semi-circle. (d) is a line tangent to ( C ) at point A.

is the perpendicular bisector of ( ∆)

[AN]. The tangent at E to ( C) cuts (d) in M and

in (∆)

U. a)

Compare the angles

and ˆ OMA

and ˆ OMA

and the angles ˆ OMU

. ˆ MOU

b) Deduce that triangle MOU is isosceles. c) Show that the circle of center U and passing through M is tangent to the line (AN). d) Show that the quadrilateral NEMO is a trapezoid. I.

Let [AB] be a diameter of a circle of center O and let M be a point on the circle. Draw at M a tangent to the circle and draw from O to the parallel to (AM) which cuts the tangent at P. a) Show that the 2 angles and are equal. ˆ POB

ˆ POM

b) Show that the two triangles OMP and OBP are equal. c) Show that (PB) is tangent to the circle.

I.

Consider circle C(O;3cm). 1) Place on circle C two points E and F such that OEF is an equilateral triangle. 2) Draw (d) the tangent to circle C at E, (d) cuts (OF) in A. 3) What is the nature of triangle OEA? 4) Show that F is the midpoint of [OA].

I.

C is a circle of center O and diameter [AB]. Let M be a point on the circle C. The parallel to (AM) passing through O cuts the tangent to the circle at M, in point I. a) Show that ˆ = IOB ˆ MOI

b) Deduce that (BI) is tangent to the circle. c) Let L be the symmetric of M with respect to I. Show that triangle MBL is right. d) The parallel at I to (MO) cuts (LB) in Q. Show that IQBO is a trapezoid. e) (MB) cuts (OI) in K. Let H be the orthogonal projection of K on [AB]. Show that the points A, M, K and H belong to the same circle . f) Show that the trapezoid IQBO is isosceles.