Chapter 20 — A Potpourri of Geometry

Problem 343. Congruent circles $O$ and $P$ are tangent, and $AP$ is tangent to circle $O$ as shown. If $AP = 15$, what is the radius of circle $P$? (MATHCOUNTS 1989)

Problem 349. What are the possible numbers of common external tangents two congruent non-concentric circles in the same plane can have? (AHSME 1952)

Problem 357. In the adjoining figure, $TP$ and $RQ$ are parallel tangents to a circle of radius $r$, with $T$ and $R$ the points of tangency. $PSQ$ is a third tangent with $S$ as point of tangency. If $TP = 4$ and $RQ = 9$, find $r$. (AHSME 1974)

Problem 366. Show that the segment from a point $P$ outside circle $O$ to the center of the circle bisects the angle formed by the two tangents from $P$ to circle $O$.

Problem 373. In the figure, $AB$, $BC$, $DC$, and $AD$ are tangent to the circle. If $AR = 3$, $\angle D = 90°$, and arc $RST$ measures $210°$, find the area of the circle. (MAΘ 1990)

Problem 374. Two circles with equal radii are drawn such that each circle passes through the center of the other. If the distance between the centers of the two circles is 6, find the area of the region common to both circles.

Problem 375. Given circle $O$ whose diameter is 12 inches long, another circle is inscribed in a quarter circle of circle $O$ as shown. Find the radius of the smaller circle. (MATHCOUNTS 1988)

Problem 380. Circle $O$ of radius 20 is inscribed in equilateral triangle $ABC$. Circle $P$ is tangent to circle $O$ and segments $AB$ and $AC$. Find the radius of circle $P$. (MATHCOUNTS 1986)

Problem 381. A hexagon is inscribed in a circle of radius $r$. Find $r$ if two sides of the hexagon are 7 units long while the other four sides are 20 units long. (USAMTS #1)

Problem 344. In the diagram, a square is built on side $AC$ of right triangle $ABC$. If $AB = 4$ and $BC = 6$, find the area of the square. (Mandelbrot #3)

Problem 345. Triangle $ABC$ is inscribed in a circle. The measures of the non-overlapping minor arcs $AB$, $BC$, and $CA$ are, respectively, $x + 40°$, $2x + 20°$, and $2x - 20°$. Find $x$.

Problem 346. The diagram shows a triangle inscribed in a circle of radius 4. Given that $BA \perp AC$ and $\angle ABC = 30°$, find the area of $\triangle ABC$. (MAΘ 1991)

Problem 347. A triangle has vertices $A(0, 15)$, $B(0, 0)$, and $C(10, 0)$. Find the coordinates of point $D$ on $AC$ so that the area of $\triangle ABD$ equals the area of $\triangle DBC$. (MATHCOUNTS 1990)

Problem 364. In a right triangle with legs $a$ and $b$ and hypotenuse $c$, the altitude drawn to the hypotenuse has length $x$. Prove that

(AHSME 1956)

Problem 377. In $\triangle ABC$, $CD$ is the altitude to $AB$ and $AE$ is the altitude to $BC$. If the lengths of $AB$, $CD$, and $AE$ are known, determine $DB$ in terms of these lengths in each of the following cases: (AHSME 1962)

i. Both angles $A$ and $B$ are acute. ii. Angle $A$ or angle $B$ is obtuse. iii. Angle $A$ is right. iv. Angle $B$ is right.

Problem 383. In triangle $ABC$, point $D$ is the midpoint of $BC$ and point $E$ is the midpoint of $AC$. If $AD = 6$, $BE = 9$, and $AC = 10$, find the area of triangle $ABC$. (Mandelbrot #2)

Problem 392. Suppose we form a triangle by connecting all the points of contact of the sides of a triangle and its incircle. Prove that the new triangle is acute. (AHSME 1954)

Problem 394. In triangle $ABC$ with medians $AE$, $BF$, and $CD$, $FH$ is parallel and equal in length to $AE$, and $FE$ extended meets $BH$ in $G$. Prove each of the following: (AHSME 1955)

i. $AEHF$ is a parallelogram. ii. $BH = DC$. iii. $FG$ is a median of $\triangle BFH$. iv. $FG = \frac{3}{4} AB$.

Problem 398. Given $\triangle ABC$ with $AB \perp BC$, $BD$ is the altitude to $AC$, $AF$ bisects $\angle BAC$, $AP = 12$, and $PF = 8$. Find $\tan \angle BAF$ and find $PD$. (MAΘ 1990)

Problem 400. Triangle $ABC$ has area 10. Points $D$, $E$, and $F$, all distinct from $A$, $B$, and $C$, are on sides $AB$, $BC$, and $CA$ respectively, and $AD = 2$, $DB = 3$. If triangle $ABE$ and quadrilateral $DBEF$ have equal areas, then what is that area? (AHSME 1983)

Problem 401. In $\triangle ABC$, $M$ is the midpoint of side $BC$, $AN$ bisects $\angle BAC$, and $BN \perp AN$. If sides $AB$ and $AC$ have lengths 14 and 19 respectively, find $MN$. (AHSME 1981)

Problem 348. Find the area of $BCDE$ if $AC = 20$, $CD = 12$, and $BE = 3$. (MATHCOUNTS 1986)

Problem 350. Quadrilateral $BDEF$ is a rhombus with vertices on $\triangle ABC$. Given $AB = 10$ and $BC = 15$, find $DE$. (MATHCOUNTS 1992)

Problem 351. Let $ABCD$ be an isosceles trapezoid such that $AB | CD$, $AB = 8$, $CD = 4$, and $AC \perp BD$. Find the area of $ABCD$. (M&IQ 1992)

Problem 354. A triangle and a trapezoid are equal in area. They also have the same altitude. If the base of the triangle is 18 inches, find the median of the trapezoid. (AHSME 1953)

Problem 356. In pentagon $ABCDE$, the perpendicular bisector of $AB$ passes through vertex $D$ and point $P$ on $AB$. The pentagon is symmetric with respect to $DP$ and $BC | AE$. If $BC = 4$, $AB = 16$, and $DP = 7$, find the area of pentagon $ABCDE$. (MATHCOUNTS 1984)

Problem 358. A square $ABCD$ has line segments drawn from vertex $B$ to the midpoints $N$ and $M$ of sides $AD$ and $DC$ respectively. Find the ratio of the perimeter of quadrilateral $BMDN$ to the perimeter of square $ABCD$. (MATHCOUNTS 1989)

Problem 359. $ABCD$ is a rectangle and $DE = DC$. Given $AD = 5$ and $BE = 3$, find $DE$. (MATHCOUNTS 1992)

Problem 372. Three sides of a quadrilateral have lengths 1, 2, and 5. The fourth side is an integer $x$. What is the sum of all possible values of $x$?

Problem 385. Let $ABCD$ be a right-angled trapezoid where $AB | CD$ and $AD \perp AB$, and let $M$ be the midpoint of $AD$. Let $BC = AB + CD$ and let $S$ be a point on $BC$ such that $CS = CD$. Prove first that $\angle ASD$ is a right angle, then use this to show that $\angle MDS = \angle MSD$. (M&IQ 1992)

Problem 391. Inside square $ABCD$, with sides of length 12 inches, segment $AE$ is drawn, where $E$ is the point on $DC$ which is 5 inches from $D$. The perpendicular bisector of $AE$ is drawn and intersects $AE$, $AD$, and $BC$ at points $M$, $P$, and $Q$ respectively. Find the ratio of segment $PM$ to $MQ$. (AHSME 1972)

Problem 396. Let $ABCD$ be an isosceles trapezoid with $AB | CD$ and $AD = BC$; let $O$ be the intersection of $AC$ and $BD$; let $\angle AOB = 60°$; and let $M$, $N$, $P$ be the midpoints of $AO$, $DO$, $BC$ respectively. Prove that triangle $MNP$ is equilateral. (M&IQ 1992)

Problem 399. Find the length of the median of a trapezoid whose diagonals are perpendicular segments of lengths 7 and 9. (Mandelbrot #3)

Problem 352. Find the area of a semicircle inscribed in $\triangle ABC$ as in the diagram, where $AB = AC = 25$ and $BC = 40$. (Mandelbrot #1)

Problem 360. Find the ratio of the perimeter of an equilateral triangle having an altitude equal to the radius of a circle to the perimeter of an equilateral triangle inscribed in the circle. (AHSME 1952)

Problem 361. In $\triangle ABC$, $\angle ACB$ is a right angle and $DE | CB$. If $AE = 3$ and $EB = 7$, find the ratio of the area of $\triangle ADE$ to the area of trapezoid $BCDE$. (MAΘ 1990)

Problem 362. An equilateral triangle is circumscribed about a circle with radius 9. Find the area of the triangle.

Problem 395. A square with side length 1 is rotated about one vertex by an angle $\alpha$, where $0° < \alpha < 90°$. If $\cos \alpha = 4/5$, find the area common to both the original square and its rotated image. (Mandelbrot #2)

Problem 355. $\triangle ABC$ is inscribed in circle $O$ with $AB = 4$, $BC = 8$, and $AC = 9$. Segment $BY$ bisects arc $AC$. Find $DC$. (MAΘ 1990)

Problem 370. Find the length of the longest diagonal of a regular dodecagon of side length 1. (Mandelbrot #3)

Problem 371. The segments from $AB$ to $F$, from $BC$ to $D$, and from $CA$ to $E$ are perpendicular bisectors of $AB$, $BC$, and $AC$. If the perimeter of $\triangle ABC$ is 35 and the radius of the circle is 8, find the area of the hexagon $AECDBF$. (Mandelbrot #1)

Problem 378. In the diagram, chords $AB$ and $CD$ intersect at point $E$ within the circle. If $CE = 12$, $AE = 8$, $AB = 14$, and $AD = 10$, find $AX$. (MAΘ 1990)

Problem 382. Circle $O$ has diameters $AB$ and $CD$ perpendicular to each other. $AM$ is any chord intersecting $CD$ at $P$. Which of the following equals $AP \cdot AM$: $AO \cdot OB$, $AO \cdot AB$, $CP \cdot CD$, $CP \cdot PD$, or $CO \cdot OP$? (AHSME 1957)

Problem 386. Let $BC$ of right triangle $ABC$ be the diameter of a circle intersecting hypotenuse $AB$ at $D$. At $D$ a tangent is drawn cutting leg $CA$ at $F$. Prove each of the following: (AHSME 1965)

i. $DF$ bisects $CA$. ii. $DF = FA$. iii. $\angle A = \angle BCD$. iv. $\angle CFD = 2\angle A$.

Problem 390. Triangle $ABC$ is inscribed in a circle with center $O$. A circle with center $I$ is inscribed in $\triangle ABC$. $AI$ is drawn and extended to intersect the larger circle at $D$. Prove that $D$ is the circumcenter of $\triangle CIB$. (AHSME 1966)

Problem 393. The two circles in the diagram are incircles of $\triangle ADB$ and $\triangle ADC$. These incircles are tangent to $AD$ and to each other at $G$. (Mandelbrot #1)

i. If $AB = c$, $AC = b$, and $BC = a$, find the length of $BD$ in terms of $a$, $b$, and $c$. ii. Let the radii of the two circles be $r$ and $s$. Show that the length of $DE$ is $\sqrt{rs}$.

Problem 353. A band is wrapped tightly around two pulleys whose centers are 6 feet apart. The radii of the pulleys are 4 feet and 1 foot. How many feet long is the band? (MAΘ 1992)

Problem 363. A corner of a rectangular piece of paper of width 8 inches is folded over so that it coincides with point $C$ on the opposite side. If $BC = 5$ inches, find the length in inches of fold $\ell$. (MATHCOUNTS 1991)

Problem 365. A square with side 6 inches is shown. If $P$ is a point such that segments $PA$, $PB$, and $PC$ are equal in length, and segment $PC$ is perpendicular to segment $ED$, what is the area in square inches of triangle $APB$? (MATHCOUNTS 1991)

Problem 367. $ABCD$ is a square and $ADCE$ is an equilateral triangle. Given $FE = 1$ and $FE | AD$, find $DC$. (MATHCOUNTS 1992)

Problem 368. In a $5 \times 12$ rectangle, one of the diagonals is drawn and then circles are inscribed in both right triangles formed. Find the distance between the centers of the two circles. (MAΘ 1987)

Problem 369. In $\triangle ABC$, $DE | BC$, $\angle B = 60°$, $AG \perp BC$, $\angle AED = 45°$, $AD = 6$, and $AB = 10$. What is the area of $FECG$? (MAΘ 1990)

Problem 376. A man is 6 miles east and 5 miles south of his home. He is also 3 miles north of a river which is 8 miles south of his home. What is the least number of miles he may travel if he must fetch water from the river, then return home? (MAΘ 1992)

Problem 379. The six edges of tetrahedron $ABCD$ measure 7, 13, 18, 27, 36, and 41 units. If the length of edge $AB$ is 41, find the length of edge $CD$. (AHSME 1988)

Problem 384. Given that $\ell | m | n$, show that $\dfrac{x}{y} = \dfrac{c}{d} = \dfrac{a}{b}$ and $\dfrac{g - f}{d} = \dfrac{h - g}{b}$, where $f$, $g$, and $h$ are the lengths of the parallel segments cut off by the two transversals as shown.

Problem 387. In the diagram, the semicircles centered at $P$ and $Q$ are tangent to each other and to the large semicircle, and their radii are 6 and 4 respectively. Line $LM$ is tangent to semicircles $P$ and $Q$. Find $LM$. (Mandelbrot #3)

Problem 388. In a narrow alley of width $w$, a ladder of length $a$ is placed with its foot at point $P$ between the walls. Resting against one wall at $Q$, a distance $k$ above the ground, the ladder makes a $45°$ angle with the ground. Resting against the other wall at $R$, a distance $h$ above the ground, the ladder makes a $75°$ angle with the ground. Find $w$ in terms of $h$ and $k$. (AHSME 1982)

Problem 389. In the diagram, points $C$ and $B$ are centers of circles that are tangent to each other. Points $E$ and $F$ are points of tangency, $BF = 4$, and $DB = 20$. Lines $DB$ and $EF$ intersect at point $A$. Find $AE$. (MAΘ 1990)

Problem 397. A large sphere is on a horizontal field on a sunny day. At a certain time the shadow of the sphere reaches out a distance of 10 m from the point where the sphere touches the ground. At the same instant, a meter stick (held vertically with one end on the ground) casts a shadow of length 2 m. What is the radius of the sphere in meters? (The answer is not $5/2$!) (AHSME 1983)