Let AB be a line drawn through the point of contact of two circles tangent to each other at the point E. Let MN and HK be tangents to the circles at the points A and B.

Ex. 120. If two circles touch each other, and two lines are drawn through the point of contact terminated by the circumferences, the chords joining the ends of these lines are parallel.

Let the touch at P, and let AB, CD be lines through P terminated by the circumferences.

To prove that AC is | to DB.

Proof. Draw through P the common tangent MN.

N

Ex. 101

§ 93

Ax. 1

§ 111

Q. E. D.

Ex. 121. If two circles intersect and a line is drawn through each point of intersection terminated by the circumferences, the chords joining the ends of these lines are parallel.

Let the intersect in the points A, B, and let CAD, EBF be any two lines through A, B terminated by the circumferences. To prove that CE is to DF.

Proof.

and

But

ACEB, ADFB are inscribed quadrilaterals.

C

D

F

Ex. 122. Through one of the points of intersection of two circles a diameter of each circle is drawn. Prove that the line joining the ends of the diameters passes through the other point of intersection.

Let O, O' be the centres of the O, A and B the points of intersection, AC, AD the two diameters.

To prove that the straight line CD passes through B.

Ex. 123. If two common external tangents or two common internal tangents are drawn to two circles, the segments intercepted between the points of contact are equal.

Let AB and CD be the two common external tangents, and EF and GH the two common internal tangents to the two circles.

Ex. 124. The diameter of the circle inscribed in a right triangle is equal to the difference between the sum of the legs and the hypotenuse.

Let ABC be a rt. ▲, ▲ B = 90°, O the centre of the inscribed O, D, E, F the points of contact.

To prove that diameter of O
Proof.

C

F

= AB+ BC AC.

E

Draw the radii OD, OE.
OD is to AB, and OE is 1 to BC.

Ex. 125. If one leg of a right triangle is the diameter of a circle, the tangent at the point where the circumference cuts the hypotenuse bisects the other leg.

Let the diameter AB of the circle ADB be the leg AB of the rt. ▲ ABC, and let the hypotenuse AC cut the circumference at D. Let DE be the tangent at D, cutting the leg BC at E.

Ex. 126. If, from any point in the circumference of a circle, a chord and a tangent are drawn, the perpendiculars dropped on them from the middle point of the subtended arc are equal.

Let the tangent AC and the chord AB be drawn from the same point A, and let M be the middle point of the arc AB.

Rt. A AHM and AKM are equal.

For MAH = Z KAM, and AM is common.

Ax. 7

§ 141

§ 128

Q. E. D.

Ex. 127. The median of a trapezoid circumscribed about a circle is equal to one fourth the perimeter of the trapezoid.

Let HK be the median of the circumscribed trapezoid ABCD.

To prove that HK = ‡ (AB + BC + CD + DA).

B

H

Ex. 103

$ 190

.. HK = 1 (AB + BC + CD + DA).

Q. E. D.

Ex. 128. Two fixed circles touch each other externally and a circle of variable radius touches both externally. Show that the difference of the distances from the centre of the variable circle to the centres of the fixed circles is constant.

Let r and r' be the radii of two fixed circles that are tangent to each other externally. Let r" be the variable radius of a circle that touches both externally. Let r be greater than r'.

To prove that the difference of the distances from the centre of the variable circle to the centres

of the fixed circles is constant.

Proof. Draw the two lines of centres.

These lines of centres pass through the points of contact.

(+) is the required difference.

Now r + r'' – (r' + r'' ) = r + p'' - p' - p'' = r — r', a constant.

§ 265

Q. E. D.

Ex. 129. If two fixed circles intersect, and circles are drawn to touch both, show that either the sum or the difference of the distances of their centres from the centres of the fixed circles is constant, according as they touch (i) one internally and one

externally, (ii) both internally or both externally.

Let O and O' be the centres and r and r' the radii of the two fixed . Let C be the centre and s the radius of the variable O tangent to one fixed internally and to the other externally. Let C' be the centre and s' the radius of the variable tangent to both the fixed internally. Let C" be

O'

the centre and s" the radius of the variable O tangent to both the fixed externally.

Draw the lines of centres OC, OC', OC", O'C', O'C', O'C".

Let r be greater than r'.

$ 265

To prove that OC+ O'C, OC′ – o'c′, oc′′ – O'C" are constant. Proof. The lines of centres pass through the points of contact. .. OC + O'C = (r − s) + (r' + s) = r − s + r′ + s = r + r', a constant; O'C — O'C' = (r− s′ ) — (r′ — s′ ) = r — s′ — r' + s′ = r - r', a constant; and OC" - O'C" = (r + s′′) − (r′ + s'' ) = r + s′′ − r — s'' = r — r', a constant.

Q. E. D.

Ex. 130. If two straight lines are drawn through any point in a diagonal of a square parallel to the sides of the square, the points where these lines meet the sides lie on the circumference of a circle whose centre is the point of intersection of the diagonals.

Let P be any point in the diagonal AC of the square ABCD, and let EE' and FF' be drawn through P || to BA and DA, respectively.

Ex. 131. If ABC is an inscribed equilateral triangle and P is any point in the arc BC, then PA

=

PB+ PC.

Proof. Upon PA take PM equal to PB, and draw BM.