Finite geometry

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Thales' theorem: if Template:Overline is a diameter, then the angle at B is a right angle.

In geometry, Thales' theorem (named after Thales of Miletus) states that if A, B and C are points on a circle where the line Template:Overline is a diameter of the circle, then the angle ∠ABC is a right angle. Thales' theorem is a special case of the inscribed angle theorem, and is mentioned and proved on the 33^rd proposition, third book of Euclid's Elements. It is generally attributed to Thales, who is said to have sacrificed an ox in honor of the discovery, but sometimes it is attributed to Pythagoras.

History

There is nothing extant of the writing of Thales; work done in ancient Greece tended to be attributed to men of wisdom without respect to all the individuals involved in any particular intellectual constructions — this is true of Pythagoras especially. Attribution did tend to occur at a later time. Reference to Thales was made by Proclus, and by Diogenes Laertius documenting Pamphila statement that the ancient Benefits of Living in a Rental or Landed property in Singapore Property New Launches & Undertaking Showcase In Singapore Many residential Singapore property gross sales contain shopping for property in Singapore at new launches. These are often properties below development, being sold new by developers. New Launch Singapore Property, 28 Imperial Residences Coming To Geylang Lorong 26 The property market is slowing down, in accordance with personal property transactions in Could Cellular Apps FREE Signal Up Log in Property Agents Feedback

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Indian and Babylonian mathematicians knew this for special cases before Thales proved it. It is believed that Thales learned that an angle inscribed in a semicircle is a right angle during his travels to Babylon. The theorem is named after Thales because he was said by ancient sources to have been the first to prove the theorem, using his own results that the base angles of an isosceles triangle are equal, and that the sum of angles in a triangle is equal to 180°.

Template:Quotebox Dante's Paradiso (canto 13, lines 101–102) refers to Thales' theorem in the course of a speech.

Proof

First proof

We use the following facts: the sum of the angles in a triangle is equal to 180° and the base angles of an isosceles triangle are equal. To achieve promoting a home, it's essential be competent in actual estate advertising, authorized, financial, operational aspects, and different knowledge and abilities. That is essential because it's essential to negotiate with increasingly subtle buyers. It is advisable outperform opponents, use latest technologies, and stay ahead of the quick changing market.

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Since Template:Overline = Template:Overline = Template:Overline, ∆OBA and ∆OBC are isosceles triangles, and by the equality of the base angles of an isosceles triangle, ∠OBC = ∠OCB and ∠BAO = ∠ABO.

Let α = ∠BAO and β = ∠OBC. The three internal angles of the ∆ABC triangle are α, (α + β), and β. Since the sum of the angles of a triangle is equal to 180°, we have

$\alpha +\left(\alpha +\beta \right)+\beta =180^{\circ }$

${2}\alpha +{2}\beta =180^{\circ }$

${2}(\alpha +\beta )=180^{\circ }$

$\therefore \alpha +\beta =90^{\circ }.$

Q.E.D.

Second proof

The theorem may also be proven using trigonometry: Let $O=(0,0)$, $A=(-1,0)$, and $C=(1,0)$. Then B is a point on the unit circle $(\cos \theta ,\sin \theta )$. We will show that ∆ABC forms a right angle by proving that Template:Overline and Template:Overline are perpendicular — that is, the product of their slopes is equal to –1. We calculate the slopes for Template:Overline and Template:Overline:

$m_{AB}={\frac {y_{B}-y_{A}}{x_{B}-x_{A}}}={\frac {\sin \theta }{\cos \theta +1}}$

and

$m_{BC}={\frac {y_{B}-y_{C}}{x_{B}-x_{C}}}={\frac {\sin \theta }{\cos \theta -1}}$

Then we show that their product equals –1:

${\begin{aligned}&m_{AB}\cdot m_{BC}\\&={\frac {\sin \theta }{\cos \theta +1}}\cdot {\frac {\sin \theta }{\cos \theta -1}}\\&={\frac {\sin ^{2}\theta }{\cos ^{2}\theta -1}}\\&={\frac {\sin ^{2}\theta }{-\sin ^{2}\theta }}\\&=-1\end{aligned}}$

Note the use of the Pythagorean trigonometric identity $\sin ^{2}\theta +\cos ^{2}\theta =1$.

Converse

The converse of Thales' theorem is also valid; it states that a right triangle's hypotenuse is a diameter of its circumcircle.

Combining Thales' theorem with its converse we get that:

The center of a triangle's circumcircle lies on one of the triangle's sides if and only if the triangle is a right triangle.

Proof of the converse using geometry

Figure for the proof of the converse

This proof consists of 'completing' the right triangle to form a rectangle and noticing that the center of that rectangle is equidistant from the vertices and so is the center of the circumscribing circle of the original triangle, it utilizes two facts:

• adjacent angles in a parallelogram are supplementary (add to 180°) and,
• the diagonals of a rectangle are equal and cross each other in their median point.

Let there be a right angle ∠ABC, r a line parallel to Template:Overline passing by A and s a line parallel to Template:Overline passing by C. Let D be the point of intersection of lines r and s (Note that it has not been proven that D lies on the circle)

The quadrilateral ABCD forms a parallelogram by construction (as opposite sides are parallel). Since in a parallelogram adjacent angles are supplementary (add to 180°) and ∠ABC is a right angle (90°) then angles ∠BAD, ∠BCD, and ∠ADC are also right (90°); consequently ABCD is a rectangle.

Let O be the point of intersection of the diagonals Template:Overline and Template:Overline. Then the point O, by the second fact above, is equidistant from A,B, and C. And so O is center of the circumscribing circle, and the hypotenuse of the triangle (Template:Overline) is a diameter of the circle.

Proof of the converse using linear algebra

This proof utilizes two facts:

• two lines form a right angle if and only if the dot product of their directional vectors is zero, and
• the square of the length of a vector is given by the dot product of the vector with itself.

Let there be a right angle ∠ABC and circle M with Template:Overline as a diameter. Let M's center lie on the origin, for easier calculation. Then we know

• A = − C, because the circle centered at the origin has Template:Overline as diameter, and
• (A − B) · (B − C) = 0, because ∠ABC is a right angle.

It follows

0 = (A − B) · (B − C) = (A − B) · (B + A) = |A|² − |B|².

Hence:

|A| = |B|.

This means that A and B are equidistant from the origin, i.e. from the center of M. Since A lies on M, so does B, and the circle M is therefore the triangle's circumcircle.

The above calculations in fact establish that both directions of Thales' theorem are valid in any inner product space.

Generalizations and related results

Thales' theorem is a special case of the following theorem:

Given three points A, B and C on a circle with center O, the angle ∠AOC is twice as large as the angle ∠ABC.

See inscribed angle, the proof of this theorem is quite similar to the proof of Thales' theorem given above.

A related result to Thales' theorem is the following:

• If Template:Overline is a diameter of a circle, then:

• If B is inside the circle, then ∠ABC > 90°
• If B is on the circle, then ∠ABC = 90°
• If B is outside the circle, then ∠ABC < 90°.

Application

Constructing a tangent using Thales' theorem.

Thales' theorem can be used to construct the tangent to a given circle that passes through a given point. (See figure.) We can refer to the image beside, this is one of the cases where Thales' Theorem is used very frequently.

See also

• Synthetic geometry

References

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• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

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• Munching on Inscribed Angles
• Thales' theorem explained With interactive animation
• Thales' Theorem by Michael Schreiber, The Wolfram Demonstrations Project.