perplexus.info :: General : Equal Tangents

Equal Tangents (Posted on 2010-10-03)

C(UV) denotes the circle with diameter UV.

T(P,QR) denotes the tangential distance |PS|,
where point P lies outside C(QR), point S
lies on C(QR), and PS is tangent to C(QR).

Let A, B, C, and D be distinct, collinear
points in that order.

Construct a point E on line AD such that
|EF| = T(E,AB) = T(E,CD) = |EG|

Submitted by Bractals

Rating: 3.0000 (1 votes)

Solution: (Hide)

Construction:

Construct circles C(AB) and C(CD). Draw a circle
K₁ intersecting circle C(AB) in points P₁ and Q₁ and
circle C(CD) in points R₁ and S₁. Let T₁ be the
intersection of lines P₁Q₁ and R₁S₁. Construct the
point T₂ similarly. The intersection of lines T₁T₂
and AD is the desired point E.

Note:

When I say construct a circle, I mean that only one
circle is possible from the information given; whereas,
when I say draw a circle, I mean that an infinite number
of circles are possible from the infomation given.

Proof:

T(X,YZ) is the square root of the "power" of point X
with respect to circle C(YZ).

The locus of points E where T(E,AB) = T(E,CD) is the
"radical line" of the circles C(AB) and C(CD).

The radical line of circles K₁ and C(AB) is the line P₁Q₁.
The radical line of circles K₁ and C(CD) is the line R₁S₁.
The point T₁ lies on the radical line of circles C(AB) and
C(CD) (see "radical center"). Similarly, T₂ lies on the
radical line of circles C(AB) and C(CD). Therefore, T₁T₂
is the radical line of circles C(AB) and C(CD).

Thus, E is the desired point.

Note:

See Larry's post for an analytic calculation of the distance
from A to E. This distance is constructible and therefore is
an alternate method for constructing point E.

Comments: ( You must be logged in to post comments.)

Subject	Author	Date
re(3): Construction	Charlie	2010-10-05 01:05:46
re: Algebraic solution leads to geometric	Larry	2010-10-04 22:34:19
re(2): Construction	brianjn	2010-10-04 21:13:39
re(2): Does this work?	broll	2010-10-04 12:20:04
re: Does this work?	Charlie	2010-10-04 12:16:49
re: Construction	Charlie	2010-10-04 12:05:22
Does this work?	broll	2010-10-04 02:53:11
Construction	brianjn	2010-10-04 02:20:35
No Subject	Ady TZIDON	2010-10-03 15:40:05
re: Algebraic solution	Bractals	2010-10-03 14:39:12
Algebraic solution	Larry	2010-10-03 14:14:53

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