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# Cách chứng minh các điểm (4 điểm) cùng thuộc một đường tròn

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So how to prove the points (4 points) belong to round like? How many ways are there to prove that 4 points are on the same circle? Let’s find out through the article below.

° Method of proving points on a circle

* Method 1: Prove that these points are equidistant from a fixed point O. Then the given points are on the same circle with center O.

* Method 2: Use inscribed quadrilaterals. For example, to prove that 5 points A, B, C, D, E belong to the same circle, we prove that ABCD, ABCE are quadrilaterals inscribed with a circle with center O.

Below, we refer to some examples illustrating how to prove that 4 points are on the same circle.

* Example 1: Let ABC be a right triangle at A, altitude AH. Since M is any point on side BC draw MD AB, ME AC. Prove that 5 points A, D, M, H, E lie on the same circle.

– According to the article, there is the following picture: Consider a right triangle ADM with hypotenuse AM

Consider a right triangle AEM with hypotenuse AM

And right triangle AHM has hypotenuse AM

These triangles all have the same hypotenuse AM, so the three vertices of the right angle lie on a circle of diameter AM whose center is the midpoint of AM.

So 5 points A, D, M, H, E lie on the same circle.

* Example 2: Let ABC be a right triangle at A, let D be the point of symmetry to A through side BC. Prove that 4 points A, B, C, D are on the same circle.

We have the following drawing: Since D is symmetric to A through BC, B is symmetric to B through BC, and C is symmetric to C over BC, so Symmetrical to the angle via BC.

Derive BDC = BAC = 900

Considering right triangle BAC and BDC have common hypotenuse BC, so two vertices of right angle A and D lie on circle with diameter BC, center is mid point of hypotenuse BC.

So 4 points A, B, C, D lie on the same circle.

* Example 3: Let ABC be a right triangle at A. On AC take point D. The projection of D on BC is E, the symmetry point of E through BD is F. Prove that 5 points A, B, E, D, F lie on the same one. circle. Find the center O of the circle.

We have the following drawing: – Hypothetically, DE BC so BEB = 900

– Since E and F are symmetrical about BD, BD is the perpendicular bisector of the line segment EF, so it follows:

BF = BE and DF = DE

Infer: ΔBFD = BED (ccc)

Infer: ∠BFD = BEB = 900

Let O be the mid point of BD.

– Consider a right triangle ABD, right angled at A with AO as the median, so:

AO = BD = OB = OD (1)

– Consider a right triangle BDE right at E with OE as the median, so:

EO = BD = OB = OD (2)

– Consider a right triangle BFD right at F with OF as the median, so:

FO = BD = OB = OD (3)

From (1), (2) and (3) deduce: OA = OB = OD = OE = OF.

So 5 points A, B, E, D, F lie on a circle with center O with O being the midpoint of BC.

Hope with the article How to prove points (4 points) are on the same circle In the 9th grade math content above, hayhochoi.vn helps students solve these types of exercises easily. If you have any suggestions and questions, please leave a comment below the article for Hay Learn to recognize and support, wish you good luck in your studies.