DOPPLER TRIANGLE

Doppler Triangle: These are very special triangles whose side lengths are determined by the signal's arrival time.
Doppler Triangles are used to analyze how the (c+v)(c-v) mathematics occurs between two reference systems, one stationary and the other in motion. The positions of the reference systems at the time of signal emission and arrival form the vertices of the triangle.

EXPLANATION:

In the figure above, we see a Doppler Triangle formed by points O, A, and B. The red line associated with the plane's reference system is an auxiliary line that we will use for analysis.

Line d₀: This is the direction of the signal's arrival relative to the observer's reference system. Since the signal traverses this line at the speed of c, the arrival time of the signal is t_Δ=d₀/c.
Line d₁: This is the distance the plane covers with speed "u" during the signal's travel time. d₁=u.t_Δ
Line d₂: This represents the direction of the signal relative to the plane's reference system.
Red Line: The red line moving with the plane represents the signal's direction relative to the plane's reference system. This line is parallel to and equal in length to the black line d₂. Thus, the black line d₂ represents the signal's direction in the plane's reference system.

As shown in the animation, the signal traverses distances AO and BO in the same time. Since d₀=c.t_Δ and d₀≠d₂, it follows that d₂=(c±v).t_Δ. Thus, the signal traverses the distance d₂ at a speed of (c±v), where ±v represents the relative speed of approach/recession between the two reference systems.

As shown, the Doppler Triangle directly leads us to the (c+v)(c-v) mathematics. The Doppler Triangle is an integral part of (c+v)(c-v) mathematics.

EXPLANATION:

As shown in this figure, the observer sends the signal.
Line d₀: The red line moving with the plane shows the direction of the signal's arrival relative to the plane's reference system. The signal traverses this red line at speed c to reach the plane. This line is parallel to and equal in length to the black line d₀. Therefore, in calculations, the black line d₀ represents the line determining the signal's arrival time.
Line d₁: This is the distance the plane covers at speed "u" during the signal's travel time.
Line d₂: Relative to the observer's reference system, this is the direction of the signal, which the signal traverses during the arrival time. Dividing the distance d₂ by the arrival time gives the speed of the signal relative to the observer's reference system.

EXPLANATION:

In this figure, the plane and the signal tower send signals to each other.

INCOMING SIGNALS: The length of the line d₀ (red line for the plane) determining the DIRECTION of the incoming signal does not change for both reference systems, and since signals traverse this line at speed c, the arrival time of the signal is the same for both reference systems.
t_Δ=d₀/c

OUTGOING SIGNALS: The length of the line d₂ (blue line for the plane) giving the DIRECTION of the outgoing signal does not change for both reference systems. Since outgoing signals reach their targets at the same time and cover the same distance, the speed of the signals sent by both reference systems is also equal.
c±v=d₂/t_Δ 

EXPLANATION:

Any of the two reference systems moving relative to each other can be considered stationary while the other is considered moving. In the animation above, in the left figure, the plane is moving, and the observer is stationary. In the right figure, the plane is stationary, and the observer is moving. We see here that the same result is obtained in both thought processes. By pressing the "Play" button, the animation shows us this equivalence.

The plane is moving, the observer is stationary
In the animation on the left, the red line moving with the plane is the direction of the signal's arrival relative to the plane's reference system. This line is parallel to and equal in length to line d₀. Therefore, line d₀ determines the signal's arrival time. The time for the signal to traverse line d₀ is t_Δ=d₀/c. Line d₂, on the other hand, is the signal's direction relative to the observer's reference system, and the signal traverses this line at speed c±v=d₂/t_Δ.

The plane is stationary, the observer is moving
In the animation on the right, we see that the signal's arrival time is again determined by line d₀. The time for the signal to traverse line d₀ is t_Δ=d₀/c. The blue line associated with the signal tower is parallel to and equal in length to line d₂, and it gives the distance the signal covers to reach the plane relative to the observer's reference system. The signal traverses this line relative to the observer at a speed of "c±v=d₂/t_Δ."

As a result, both thought processes yield the same result.

VALUE OF v IN THE DOPPLER TRIANGLE