ТОП-25 фактов по геометрии для №16 ЕГЭ 2022. Доказывать на экзамене, или нет? Одобрено экспертом ✔

Introduction

The speaker introduces the video and explains that they will be discussing important geometric facts, theorems, and constructions that are commonly used in math exams. They also clarify whether or not these concepts can be used without proof on an exam.

Theorem of Menelaus

The speaker discusses the Theorem of Menelaus, which states that if a line intersects two sides of a triangle, then it also intersects the third side in such a way that the product of the three resulting segments is equal to one. This theorem can be used without proof on an exam.

• The theorem allows for finding ratios between segments on a triangle.

• It is useful for solving problems in stereometry.

• No proof is needed to use this theorem on an exam.

Theorem of Ceva

The speaker discusses the Theorem of Ceva, which states that if three lines are drawn from each vertex of a triangle to intersect with opposite sides, then their corresponding segments are proportional. This theorem can also be used without proof on an exam.

• This theorem allows for finding ratios between segments on a triangle.

• It is useful for solving problems in stereometry.

• No proof is needed to use this theorem on an exam.

Other Important Geometric Facts

In this section, the speaker covers other important geometric facts and constructions that may be useful for solving math problems.

Theorem of Van Aubel

• States that if points D and E lie outside triangle ABC such that AD, BE, and CF are concurrent, then: (AD/DB + AE/EC) = AF/FC.

• This theorem can be used without proof on an exam.

Other Theorems

• The speaker mentions that there are many other theorems that may be useful for solving math problems.

• Whether or not to use a proof depends on the specific problem being solved.

Recap and Conclusion

The speaker summarizes the main points covered in the video and emphasizes the importance of understanding these geometric concepts for success on math exams.

• Understanding important geometric facts, theorems, and constructions is crucial for success on math exams.

• It is important to know which concepts can be used without proof on an exam.

• The speaker encourages viewers to practice applying these concepts to different types of problems.

Introduction to Geometry Theorems

In this section, the speaker introduces three powerful geometry theorems that can be used in solving problems related to triangles and polygons.

Three Powerful Geometry Theorems

• The three geometry theorems are very useful for solving problems related to triangles and polygons.

• These theorems have been verified by an expert commission and are widely accepted in mathematics.

• The first theorem is called "Tallinn's theorem" which states that if a quadrilateral is inscribed in a circle with equal sides, then the product of its diagonals is equal to the sum of the products of opposite sides.

• The second theorem is known as "Heron's formula" which can be used to calculate the area of a triangle given its side lengths.

• The third theorem involves finding the length of medians or bisectors in a triangle using only its side lengths.

Trapezoid Properties

• A trapezoid has a special property where if you extend its legs and draw diagonals, they will intersect at a point that divides one base into two equal parts.

Bisectors and Circles

In this section, the speaker discusses bisectors and circles in geometry.

Bisectors of a Triangle

• The angle between the bisector of angles A and C is 90 degrees plus B divided by two.

• The fact that the angle between the bisector of angles A and C is 90 degrees plus B divided by two can be easily proven using the sum of angles in a triangle. This fact should be demonstrated during exams.

• If we draw external bisectors from angles A and C, they intersect at point L, which is also the center of an exscribed circle around triangle ABC. This fact can be used without proof but it's important to prove that angle A or C equals 90 minus B over two.

Inscribed Circle

• An inscribed circle is a circle that touches sides AB, BC, and AC externally. It lies outside the triangle with its center at point L where external bisectors from angles A and C intersect. This fact can be used without proof but it's important to prove that angle A or C equals 90 minus B over two.

Trapezoid Properties

• In a trapezoid ABCD with bases AB and CD, if the sum of angles at vertices A and D adjacent to base CD equals 90 degrees then MN = (AB - CD)/2 where M is midpoint of AB and N is midpoint of CD. This fact can be easily proven by extending sides AD and BC until they meet at point E forming right triangles AME and DNE with hypotenuse DE which has length (AB - CD).

Bisectors of a Quadrilateral

• In a quadrilateral ABCD, if BB1 is the bisector of angle B and intersects sides AC and AD at points X and Y respectively, then CX/AX = CY/DY. This fact can be used without proof.

Area of a Quadrilateral

• The area of any convex quadrilateral ABCD with diagonals AC and BD is equal to half the product of the lengths of its diagonals multiplied by the sine of the angle between them. This fact can be used without proof.

Tangent Chord Angle

• If AB is a chord in a circle with center O and tangent line through A intersects chord AB at point C, then angle AOC equals 90 degrees. This fact can be used without proof.

Theorem of Sines

In this section, the speaker explains how to apply the theorem of sines when given a circle and a chord.

Applying the Theorem of Sines with a Circle and Chord

• If we have only one circle tangent to a chord and we know the length of the chord and angle between it and the tangent, we can apply the theorem of sines.

• If alpha is equal to angle AMB, where M is midpoint of AB, then sin(alpha)/2 = radius/AB.

• When there is no point M but only chords AB and CD are known along with angle between them, we can still use theorem of sines.

Geometric Constructions

In this section, the speaker discusses various geometric constructions that can be used in problem-solving.

Bisectors Construction

• If an external bisector is drawn for an angle in triangle ABC, it intersects internal bisectors at point K.

• Point K lies on circumcircle around triangle ABC.

Midpoint Construction

• If a perpendicular line is drawn from center O to chord AB in circle O, it will divide AB into two equal parts.

• Conversely, if midpoint N is taken as center for circle O', then ON will be perpendicular to AB.

Parallelogram Construction

• Joining midpoints P,Q,R,S of quadrilateral ABCD forms parallelogram PQRS.

Lemma A Treugolnikov (A Trisect)

In this section, the speaker introduces Lemma A Treugolnikov (A Trisect), which states that four segments are equal in certain conditions.

Statement of Lemma A Treugolnikov (A Trisect)

• Given triangle ABC with circumcircle and bisector of angle B intersecting the circle at point P.

• Let I be the center of the inscribed circle in triangle BC, and let K be the intersection point of external bisectors of angles A and C.

• Then segments PA, PI, PC, and BK are equal.

Applying Lemma A Treugolnikov (A Trisect)

In this section, the speaker discusses how to apply Lemma A Treugolnikov (A Trisect) in problem-solving.

Using Lemma A Treugolnikov (A Trisect)

• If a problem involves proving a part of Lemma A Treugolnikov (A Trisect), it is acceptable to use it without proof.

• However, if a problem requires proving the entire lemma, then it must be proven.

Importance of Understanding Geometric Constructions

In this section, the speaker emphasizes the importance of understanding geometric constructions and their applications in solving problems on exams like the EGЭ.

Understanding the Lemniscate Trefoil Construction

• The Lemniscate Trefoil construction is an important concept to understand for solving geometry problems on exams like the EGЭ.

• It's important to know how to apply this construction and its associated proofs in order to avoid getting a zero score on exam questions that require it.

The Importance of Knowing Orthocenter Properties

• Knowing properties related to orthocenters is also crucial for success on geometry exams.

• These properties can be used to solve a variety of problems, including those related to inscribed quadrilaterals and circles.

Using Trigonometry in Geometry Problems

• Trigonometry can also be useful in solving geometry problems, particularly when dealing with triangles.

• By using trigonometric formulas for finding areas of triangles, it's possible to simplify complex calculations and arrive at solutions more quickly.

Overall, understanding key geometric constructions, orthocenter properties, and trigonometric formulas can greatly improve one's ability to solve geometry problems on exams like the EGЭ.

Introduction to Radical Axis

In this section, the speaker introduces the concept of radical axis and explains how it can be used in geometry problems.

Definition of Radical Axis

• The radical axis is a line that passes through the intersection points of two intersecting circles.

• It is also known as the power line or radical line.

Construction of Radical Axis

• To construct the radical axis, draw a line passing through the intersection points of two intersecting circles.

• This line is called the radical axis because it has equal power with respect to both circles.

Geometric Fact about Radical Axis

• If a point P lies on the radical axis, then its distances from each circle are equal.

• This fact can be proved using basic school-level knowledge about tangents and secants.

Proof of Geometric Fact about Radical Axis

In this section, the speaker provides a proof for the geometric fact introduced earlier regarding points lying on the radical axis.

Proof of Geometric Fact

• Let PQ and PK be tangents drawn from point P to circles A and B respectively.

• Then, PQ^2 = PA * PB and PK^2 = PA * PB (using tangent properties).

• Therefore, PQ^2 = PK^2 which implies that PQ = PK.

• Hence, if P lies on the radical axis, then its distances from each circle are equal.

Formulas for Chords in Triangles

In this section, the speaker introduces formulas for finding chord lengths in triangles using Chevian's theorem.

Chevian's Theorem

• Chevian's theorem states that if a cevian intersects a side of a triangle into segments x and y, then:

• (x/a) + (y/b) = 1, where a and b are the lengths of the other two sides.

• A cevian is a line segment that connects a vertex of a triangle with the opposite side.

Formula for Chords in Triangles

• Let ABC be a triangle and let D be a point on side BC such that AD is a cevian.

• Let E and F be points on AB and AC respectively such that DE is parallel to AB and DF is parallel to AC.

• Then, using Chevian's theorem, we can derive the formula:

• BD/DC = (AB/AC) * (AF/BE)

• This formula can be used to find chord lengths in triangles.

Introduction to Proving Mathematical Theorems

In this section, the speaker introduces the concept of proving mathematical theorems and explains that it involves demonstrating a known fact or theorem.

Understanding the Task at Hand

• To prove a mathematical theorem, one must demonstrate a known fact or theorem.

• The speaker has covered most of the facts that need to be proven and has addressed questions about what needs to be proven.

Conclusion and Call-to-Action

• The speaker encourages viewers to like, subscribe, and comment on their videos.

• Viewers are asked for feedback on what topics they would like to see in future videos.

• The video ends with music playing in the background.