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MATH / In the diagram, four circles of radius 1 with centres P, Q, R, and S are tangent to one another and to the sides of △ABC, as shown. [asy] size(250); pair A, B, C, P, Q, R, S; R=(0,…
Problem
Answer published by the source. Consult the official source to check your work against its answer.
problem
In the diagram, four circles of radius 1 with centres , , , and are tangent to one another and to the sides of , as shown. [asy]
size(250);
pair A, B, C, P, Q, R, S;
R=(0,0);
Q=(-2,0);
S=(2,0);
P=(1,1.732);
B=(-5.73,-1);
C=(3.732,-1);
A=(1.366,3.098);
draw(A--B--C--A);
draw(circle(P, 1));
draw(circle(Q, 1));
draw(circle(R, 1));
draw(circle(S, 1));
label("A", A, N);
label("B", B, SW);
label("C", C, SE);
dot(P);
dot(Q);
dot(R);
dot(S);
label("P", P, N);
label("Q", Q, SW);
label("R", R, SW);
label("S", S, SE);
[/asy]
The radius of the circle with center is decreased so that
the circle with center remains tangent to ,
the circle with center remains tangent to the other three circles, and
the circle with center becomes tangent to the other three circles.
The radii and tangencies of the other three circles stay the same. This changes the size and shape of . is the new radius of the circle with center . is of the form . Find .
Plain-text mathematical notation (without MathML)
In the diagram, four circles of radius 1 with centres P, Q, R, and S are tangent to one another and to the sides of △ABC, as shown. [asy]
size(250);
pair A, B, C, P, Q, R, S;
R=(0,0);
Q=(-2,0);
S=(2,0);
P=(1,1.732);
B=(-5.73,-1);
C=(3.732,-1);
A=(1.366,3.098);
draw(A--B--C--A);
draw(circle(P, 1));
draw(circle(Q, 1));
draw(circle(R, 1));
draw(circle(S, 1));
label("A", A, N);
label("B", B, SW);
label("C", C, SE);
dot(P);
dot(Q);
dot(R);
dot(S);
label("P", P, N);
label("Q", Q, SW);
label("R", R, SW);
label("S", S, SE);
[/asy]
The radius of the circle with center R is decreased so that
• the circle with center R remains tangent to BC,
• the circle with center R remains tangent to the other three circles, and
• the circle with center P becomes tangent to the other three circles.
The radii and tangencies of the other three circles stay the same. This changes the size and shape of △ABC. r is the new radius of the circle with center R. r is of the form (a+√(b))/(c). Find a+b+c.Original LaTeX notation
In the diagram, four circles of radius 1 with centres $P$, $Q$, $R$, and $S$ are tangent to one another and to the sides of $\triangle ABC$, as shown. [asy]
size(250);
pair A, B, C, P, Q, R, S;
R=(0,0);
Q=(-2,0);
S=(2,0);
P=(1,1.732);
B=(-5.73,-1);
C=(3.732,-1);
A=(1.366,3.098);
draw(A--B--C--A);
draw(circle(P, 1));
draw(circle(Q, 1));
draw(circle(R, 1));
draw(circle(S, 1));
label("A", A, N);
label("B", B, SW);
label("C", C, SE);
dot(P);
dot(Q);
dot(R);
dot(S);
label("P", P, N);
label("Q", Q, SW);
label("R", R, SW);
label("S", S, SE);
[/asy]
The radius of the circle with center $R$ is decreased so that
$\bullet$ the circle with center $R$ remains tangent to $BC$,
$\bullet$ the circle with center $R$ remains tangent to the other three circles, and
$\bullet$ the circle with center $P$ becomes tangent to the other three circles.
The radii and tangencies of the other three circles stay the same. This changes the size and shape of $\triangle ABC$. $r$ is the new radius of the circle with center $R$. $r$ is of the form $\frac{a+\sqrt{b}}{c}$. Find $a+b+c$.level
Level 5
type
Geometry
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