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Reversal of the Thale Theorem

If one draws a series of right triangles with the same hypotenuse AB(average), it is easy to obtain the conjecture that all vertices C lie on a circle - pay someone to do my homework . This correct conjecture represents the inverse of the theorem of the Thale, which is obtained by interchanging the premise and the assertion.

If a triangle ABC is right-angled with the right angle at C, then C lies on the circle with the diameter AB(average). This circle is also called a Thales circle.

Applications of the Thales theorem in solving construction problems

If a line AB(average) is said to appear from a point P at an angle γ - pay someone to do my math homework , this means the angle BPA.

Reversing the theorem of the Thale yields:

The set of all points from which a distance AB(average) appears at an angle 90° from is a circle with AB(average) as its diameter. The Thales circle can be used as a line of determination for points that are vertices of a right angle.


Of a triangle, c, hc and γ=90° are given.

Construction description:

Points A and B are determined by c.

The point C lies:

1. on the Thales circle above AB(average) and

2. on a parallel to AB(average) at a distance hc.


The tangents to a circle k through a point P outside the circle are to be constructed.

Preliminary remarks:

The points of contact X1 and X2 of the tangents are sought - geometry homework help . It is ∢ PX1 M=90° or ∢ MX2 P=90°, since the tangent and the radius of contact are perpendicular to each other.

Construction description:

The point X lies:

1. on the circle k and

2. on the Thales circle with diameter PM(average).

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