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. В треугольнике АВС на стороне ВС отмечены точки ...

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問題

. В треугольнике АВС на стороне ВС отмечены точки N и Мтак, что BN:NM:MC=1:2:4. На стороне АС отмечена точка Р так, что AP:PC=3:2. Отрезки AN и ВР пересекаются в точке К, а отрезки AM и ВР - в точке Т. Найдите площадь треугольника АКТ, если площадь треугольника АВС равна 1035.

Python

python
import sympy as sp

# Define points
B = sp.Point(0,0)
C = sp.Point(7,0)
h = sp.symbols('h', positive=True)
A = sp.Point(0,h)
N = sp.Point(1,0)
M = sp.Point(3,0)
P = sp.Point(21/5, 2*h/5)

# Lines
line_BP = sp.Line(B,P)
line_AN = sp.Line(A,N)
line_AM = sp.Line(A,M)

K = line_BP.intersection(line_AN)[0]
T = line_BP.intersection(line_AM)[0]
A, K, T


(Point2D(0, h), Point2D(21/23, 2*h/23), Point2D(7/3, 2*h/9))

Python

python
def triangle_area(P1,P2,P3):
    return abs(sp.Polygon(P1,P2,P3).area)

area_AKT = triangle_area(A,K,T)
area_ABC = triangle_area(A,B,C)
sp.simplify(area_AKT / area_ABC), area_AKT


(14/69, 49*h/69)

Python

python
from fractions import Fraction
Fraction(14,69) * 1035


Fraction(210, 1)

思考

Area solution logic

We calculated the area as 210, which is nice and neat. To solve the triangle’s area, we can use an intercept theorem or work directly with coordinates. I set up a coordinate system and determined the equations for lines BP, AN, and AM. Solving intersecting equations reveals the point P and its related area calculations.

Calculating area ratio

We solve for the intersection of line AM with line BP, leading to x = 7/3 and y = 2h/9. We can compute the area of the triangle using coordinates, and derive the area of triangle AKT by using a ratio of areas, arriving at a final result of 14/69 for the area ratio between AKT and ABC. It all checks out!

Finalizing reasoning

We confirmed that area(AKT) = 210 by using the area ratio method. We also verified the coordinates for B, C, and A, and calculated the intersection points with line equations. For simplicity, we used the ratio method to avoid dealing with h, and presented a clean solution with all steps detailed in Russian.

Simplifying the solution steps

To find the area of triangle AKT, we first solve for the intersection points K and T by using line equations. Then, we can compute the area using the polygon area formula, but to make it simpler, we mention the area is proportional to h. The area of AKT works out to be 49h/69, which simplifies the process!

Thought for 1m 2s

回答

Ответ: 210

Короткое решение через координаты:

  1. Положим B=(0,0), C=(7,0)B=(0,0),\,C=(7,0) (так удобно, т.к. BN:NM:MC=1:2:4BN:NM:MC=1:2:4), а A=(0,h)A=(0,h). Тогда
    N=(1,0)N=(1,0), M=(3,0)M=(3,0).

  2. Точка PP на ACAC так, что AP:PC=3:2AP:PC=3:2. Значит

P=2A+3C5=(215, 2h5).P=\frac{2A+3C}{5}=\Big(\frac{21}{5},\,\frac{2h}{5}\Big).

Прямая BPBP: y=2h21xy=\frac{2h}{21}x.

  1. Прямые AN:y=h(1−x)AN: y=h(1-x) и AM:y=h ⁣(1−x3)AM: y=h\!\left(1-\frac{x}{3}\right). Пересечения с BPBP:
K:{h(1−x)=2h21x⇒xK=2123, yK=2h23;T:{h ⁣(1−x3)=2h21x⇒xT=73, yT=2h9.K: \begin{cases} h(1-x)=\frac{2h}{21}x \end{cases}\Rightarrow x_K=\frac{21}{23},\ y_K=\frac{2h}{23}; \quad T: \begin{cases} h\!\left(1-\frac{x}{3}\right)=\frac{2h}{21}x \end{cases}\Rightarrow x_T=\frac{7}{3},\ y_T=\frac{2h}{9}.
  1. Площадь △AKT=12∣ AK→×AT→ ∣\triangle AKT=\dfrac12|\,\overrightarrow{AK}\times\overrightarrow{AT}\,|:
SAKT=12 ∣xK(yT−h)−xT(yK−h)∣=12 h(9869)=49h69.S_{AKT}=\frac12\,\Big|x_K\big(y_T-h\big)-x_T\big(y_K-h\big)\Big| =\frac12\,h\left(\frac{98}{69}\right)=\frac{49h}{69}.
  1. Площадь всего △ABC=12⋅BC⋅h=72h\triangle ABC=\dfrac12\cdot BC\cdot h=\dfrac72 h.
    Значит
SAKTSABC=49h/697h/2=1469⇒SAKT=1469⋅1035=210.\frac{S_{AKT}}{S_{ABC}}=\frac{49h/69}{7h/2}=\frac{14}{69} \quad\Rightarrow\quad S_{AKT}=\frac{14}{69}\cdot 1035=210.

Ответ: 210.

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