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A DIFFERENT DEMONSTRATION OF
THE PYTHAGOREAN THEOREM USING THE POWER OF THE POINT WITH RESPECT TO
In the present paper I will give / offer you not only a different demonstration of the Pythagorean Theorem but also some demonstrations of other important theorems in geometry (The Height Theorem, The Catheters Theorem and The Generalized Pythagorean Theorem), using The Power of the Point with Respect to a Circle.
I would also like to add that these demonstrations are completely different from those already known.
THE
POWER OF A POINT WITH RESPECT TO
In this section I will point out / bring to mind what the power of a point with respect to a circle is.
STATEMENT 1. Any point P inside a circle and chord AB which contains this point, the product
PA ∙ PB is constant.
Demonstration. Given a point P inside a circle of centre O and ray / radius R and the variable chord AB, so that P _{} (AB). We build / draw a chord CD through point P, see figure above.
Angles _{}and _{} have equal measures with half measure of the arc _{}, we obtain
_{} (1)
Angles APC and BPD being vertically opposed angles, are congruent.
_{} (2)
From (1) and (2), we obtain: _{},
we obtain / get the proportion _{}, which is equivalent to the
statement _{}.
The constant value of the product _{} multiplied by (1) is called the power of an internal point P with respect to the circle.
STATEMENT 2. Any point P outside a circle and the secant which contains this point and intersects the circle in points A and B (A_{}(PB)), the product
_{} is constant.
Demonstration. Given a point P outside the circle of centre O and ray / radius R and the secant PA which intersect the circle in A and B. We build a secant PC through point P, which intersects the circle in C and D, see figure above.
From m(_{}) + m(_{}) = _{}
and
m(_{}) + m(_{}) = _{},
we deduce that
m(_{}) = m(_{}), resulting in
_{}
We have
_{} and
_{}, _{}
_{}
From the similarity of the two triangles we obtain the proposition:
_{}
i.e. _{}.
The constant value of the product _{} is called the power of an external point P with respect to the circle.
Obs. If / As _{}, then the straight line P is tangent to the circle and
_{} becomes
_{}
2. THE PYTHAGOREAN THEOREM
If in a triangle ABC, right angled in A, then _{} .
Demonstration. We consider the triangle ABC a rightangled one, inscribed in the circle with the center O and radius R. Through O, the middle of the hypotenuse BC, we draw the parallel line OD for AB, D_{}(AC) which intersects the circle in the points E and F; D_{}(OE). As [OD] is a midline in the triangle ABC, we deduce that _{}. Considering the power of the interior point D towards the circle, we have DA. _{}, resulting that _{}(1)
Taking into account that _{} and that _{}, the equality (1) becomes _{}, and we obtain _{}.
* This demonstration is different from the ones we know and it appeared for the first time in the magazine Scoala Valceana The Mathematics notebook of the students in Valcea edited by The School Inspectorate of Valcea County, The Society for Mathematical Sciences Rm. Valcea subsidiary, Teachers Assembly house in Valcea, year 19
THE RECIPROCAL PYTHAGOREAN THEOREM
If we have _{} in a triangle ABC, then the angle A is a right one.
Demonstration. We consider the triangle ABC in which _{}, inscribed in the circle with the center O and radius R. Through M, the middle of the side BC, we draw the parallel line MD for AB, _{} which intersects the circle in E and F; _{}. As [MD] is midline in the triangle ABC, we deduce that _{}. Considering the power of the interior points M and D towards the circle, we have _{} (1) and _{} respectively (2).
The equality no. (2) becomes _{}, resulting in _{} (3)
Taking into account that _{}, the equality no. (3) turns into _{}
then in _{}. (4)
From _{} and (4) we get _{} and (1) leads to _{}.
From _{} and the fact that the points E, B, C belong to the circle with the center O and radius R, we deduce that the point M is identical with the center O and that [BC] is the diameter of the circle, which means that _{}, that is the triangle ABC is rightangled in A.
THE ALTITUDE THEOREM
If [AD] is the altitude within the triangle ABC which is rightangled in A, then _{}.
Demonstration. The triangle ABV which is rightangled in A is inscribed in the circle with the center O and diameter BC. We draw _{}, _{} and note as E the point of the intersection between the straight line AD and the circle; see the figure! As [BC] is a diameter perpendicular to the chord AE in the point D, we deduce that _{}. According to the power of the interior point D towards the circle, we have _{}(2)
Taking (1) into account, the equality (2) becomes _{}.
THE CATHETUS THEOREM
If in the triangle ABC which is rightangled in A, _{} A, then _{}.
Demonstration. The triangle ABC being rightangled in A is inscribed in the circle with the center O and diameter AC; _{}. As the triangle ABC is rightangled in A, the straight line BA is tangent to the circle in the point A. According to the power of the point B which is external to the circle, we have _{}, that is _{}.
Notice In parallel, it shall be demonstrated that_{}.
PYTHAGOREAN THEOREM GENERALISED
I. If the triangle ABC has the acute angle B and _{}A, then _{}.
Demonstration. The triangle ADC being rightangled in D is inscribed in the circle with the center O and diameter AC; _{} The straight line BO intersects the circle in the points E and F; _{}; see the figure!
According to the power of the point B which is external to the circle, we have _{}, resulting in _{}, which turns into _{}, then into
_{} (1)
As [BO] is a median line in the triangle ABC and its length is given by _{}, then the equality no. (1) becomes
_{},
leading us to _{}.
II. If the triangle ABC has the obtuse angle B and _{}A, then _{}
Demonstration. The triangle ABD being rightangled in D is inscribed in the circle with the center O and diameter AB; _{}. The straight line CO intersects the circle in the points E and F; _{}; see the figure!
According to the power of the point C which is external to the circle, we have _{}, leading to _{}, that is _{} (1)
Knowing that [CO] is a median line in the triangle ABC and its length is given by _{}, the equality no.(1) becomes
_{},
resulting in
_{}
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