A circle touches the lines OA, OB and AB where OA and OB are perpendicular. Show that the diameter of the circle is equal to the perimeter of the triangle
A small circle fits between two touching circles so that all three
circles touch each other and have a common tangent? What is the
exact radius of the smallest circle?
Ten squares form regular rings either with adjacent or opposite
vertices touching. Calculate the inner and outer radii of the rings
that surround the squares.
A proof of this result came from students from
the Key Stage 3 Maths Club at Strabane Grammar School, Northern
Ireland and the following one from Joel, ACS (Barker)
(a) In the diagram given (in 2 dimensions) p
2 + q 2 = r 2 +
Draw 2 lines through V parallel to the edges of the rectangle,
dividing the figure into four pairs of right-angled triangles.
Label the point on AB as E, on BC as F, on CD as G and on DA as H.
By Pythagoras theorem:
Since DG=AE, CG=BE, AH=BF and CF=DH,
p 2 + q 2 = AE
2 + CF 2 + CG 2 + AH 2
= r 2 + s 2
(b) If the diagram represents a pyramid on a rectangular base
where p, q, r and s are the lengths of the sloping edges then the
result p 2 + q 2 =
r 2 + s 2 still holds
Let V 1 be the foot of the perpendicular from V to
the base ABCD of the pyramid and let h be the height of the pyramid
so that VV 1 = h and let V 1 A = s
1 , V 1 B = q 1 , V 1 C
= r 1 , and V 1 D= p 1 .
By Pythagoras theorem we have: p 1 2 + h
2 = p 2 , q 1
2 + h 2 = q 2 , r
1 2 + h 2 = r
2 and s 1 2 + h 2 =
s 2 .
Using the result already proved in 2dimensions, that is
p 1 2 + q 1 2 = r
1 2 + s 1 2 ,
we get p 1 2 + q 1 2
+ 2h 2 = r 1 2 + s 1
2 + 2h 2
so p 2 + q 2 =
r 2 + s 2 .