Question

In the rectangle ABCD, AB = 6 and BC = 8. The diagonals AC and BD intersect at O. Point P lies on the diagonal AC such that AP = 1. A line is drawn from B through P and meets AD at S. Let be R a point on AD such that OR is parallel to BS. a) Find the lengths of AS and RD. Hint: Denote AS = x. Use P S k OR and OR k BS to find AR and RD in terms of x. Then, find the value of x. b) Find the areas of 4AOD and 4COD. c) Find the areas of 4AOS and 4AP S. Hint: Consider that 4AOS and 4AOD have the same altitude. Use 4AP S ∼ 4AOR

let k= area and 4=triangle

Answer #1

Let ABCD be a rectangle with AB = 4 and BC = 1. Denote by M the
midpoint of line segment AD and by P the leg of the perpendicular
from B onto CM.
a) Find the lengths of P B and PM.
b) Find the area of ABPM.
c) Consider now ABCD being a parallelogram. Denote by M the
midpoint of side AD and by P the leg of the perpendicular from B
onto CM. Prove that AP =...

Let ABCD be a cyclic quadrilateral, and let the diagonals meet
in point S. From S, draw perpendiculars to the four sides, with
feet L on AB, M on BC, N on CD and K on DA. This gives a new
quadrilateral LM N K. Prove that the sum of two opposite angles in
this new quadrilateral is double the corresponding angle at the
intersection S. (E.g.∠L+∠N= 2∠ASB). Hint: Find four smaller cyclic
quadrilaterals in the diagram.

ABC is a right-angled triangle with right angle at A, and AB
> AC. Let D be the midpoint of the side BC, and let L be the
bisector of the right angle at A. Draw a perpendicular line to BC
at D, which meets the line L at point E. Prove that
(a) AD=DE; and
(b) ∠DAE=1/2(∠C−∠B)
Hint: Draw a line from A perpendicular to BC, which meets BC in
the point F

5. Suppose that the incenter I of ABC is on the triangle’s Euler
line. Show that the triangle is isosceles.
6. Suppose that three circles of equal radius pass through a
common point P, and denote by A, B, and C the three other points
where some two of these circles cross. Show that the unique circle
passing through A, B, and C has the same radius as the original
three circles.
7. Suppose A, B, and C are distinct...

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