Question

Trapezoid ABCD has ∠BAD = ∠ADC = 90◦ , AB = x, and DC = y with x < y. Diagonals AC and BD intersect at X. Point L is on AD and point M is on BC so that LM is parallel to AB and LM passes through X. Determine the length of LM in terms of x and y.

Answer #1

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...

Consider a quadrilateral ABCD such that ∠BAD and ∠ADC are
perpendicular, the rays AB and CD are on the same side of the line
AD, and AB ≅ CD.
Prove the following claims on E2, H2, and
S2.
∠ABC ≅ ∠DCB.
the perpendicular bisector of AD is also the perpendicular
bisector of BC.
Hint: Look for symmetries.

Consider a quadrilateral ABCD such that ∠BAD and ∠ADC are
perpendicular, the rays AB and CD are on the same side of the line
AD, and AB ≅ CD. Quadrilaterals with these properties are called
Khayyam quadrilaterals
Prove the following claims on E2, H2, and
S2.
∠ABC ≅ ∠DCB.
the perpendicular bisector of AD is also the perpendicular
bisector of BC.
Hint: Look for symmetries.

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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