A method to determine the heart of a proof triangle
In a triangle, three bisectors of three internal angles intersect at one point, which is the center of the inscribed circle of the triangle, also called the heart of the triangle. The distance from the center of the triangle to the three sides is equal.
The center of gravity, outer center, hanging center, inner center and lateral center of a triangle are called the five centers of the triangle. The five-center theorem of triangle refers to the triangle's center of gravity theorem, outer center theorem, vertical center theorem, inner center theorem and near center theorem.
The bisectors of angles ∠B and ∠C intersect at AC, and AB intersects at F and D..
CD and BF cross I, AI cross BC, and extend to E.
According to Seva's theorem:
BF and CD are bisectors of angles.
The theorem of angular bisector includes:
Is there an angular bisector of AE ∠A from the inverse theorem of the angular bisector theorem?
Prove the definition of judgment in triangle
A property of angular bisector: the two sides of angular bisector are proportional to the two sides of the angle.
In △ABC, BO and AC are connected at E, and O is the inner core, so BE is the bisector of ∠B, and AD passes through the inner core O (these are known from the definition of the inner core), so in △ADB, BO is the bisector of ∠B, so AB/BD=AO/OD.
Similarly, AO/OD=AC/CD.
Heart: The intersection of bisectors of three angles of a triangle is also the center of the inscribed circle.
Theorem proof used in this problem
In △abc, AD is the bisector of △ A, D is on bc, abc is the opposite side of the angle, and d=AD. Since sine theorem b/sinb = c/sind = r 1 sinb = r2sinc, is r1half the circumscribed circle of △ABD? Diameter, R2 is the radius of the circumscribed circle of △ACD, so r1/R2 = sinc/sinb = c/B. BD=R 1sinBAD, CD=R2sinCAD, ∠CAD=∠BAD, so BD/CD = r6544.
Prove the inner judgment property of triangle
Let the inscribed circle of △ABC be ☉I(r), and the opposite sides of A, B and C are A, B, C and p=(a+b+c)/2 respectively.
1, the distance from the center of the triangle to the three sides is equal, which is equal to the radius r of the inscribed circle.
2、∠BIC=90 +∠BAC/2
3. In rtδABC ∠ A = 90, and the inscribed circle of the triangle cuts BC in D, then SδABC = BD×CD.
4. The necessary and sufficient conditions for point O to be any point on the plane ABC and point I to be △ABC are: vector OI=[a (vector OA)+b (vector OB)+c (vector OC)]/(a+b+c).
5. In △ABC, the internal coordinates are:
6. (euler theorem) △ABC, where R and R are the radii of the circumscribed circle and the inscribed circle respectively, and the distance from the outer center to the inner center is d, then d? =R^2-2Rr
7. In △a, B and C: A, B and C are three sides respectively, and S is the area of a triangle, then the radius of the inscribed circle is r=2S/(a+b+c).
inscribed circle
8. The projection of the center of a triangle consisting of a point and two focal points on any branch of a hyperbola on the real axis is the vertex of the corresponding branch.
9. In △ABC, if the inscribed circle is tangent to AB, BC and CA at p, q and r respectively, then AP=AR=(b+c-a)/2, BP = BQ = (a+c-b)/2, and Cr = CQ = (b+a-c)/2.
10. Theorem of bisector of triangle interior angle: In △ABC, I is the center, and bisectors of interior angles of ∠BAC, ∠ABC and ∠ACB intersect BC, AC and AB in A', B' and C' respectively, then Ba'/CA'= AB/AC, AB.
Articles related to proving the internal judgment method of triangle;
★ Review the knowledge points of triangle proof in eighth grade mathematics.
★ Mathematics review for the senior high school entrance examination: round test center, triangular test center.
★ Summary of isosceles triangle knowledge
★ Math Triangle Review Test Questions and Answers for Senior High School Entrance Examination
★ Summary of knowledge points in the first volume of mathematics in grade three.
★ Knowledge points of Unit 1 of Mathematics in Grade Three (summer preview)
★ Reflections on the teaching of isosceles triangle judgment in eighth grade mathematics
★ Summarize the knowledge points of mathematics in the first volume of Grade Three.
★ Junior high school ninth grade mathematics knowledge points
★ Senior high school entrance examination mathematics high frequency test center