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What is the difference between "hamstring tendon" and "common hamstring tendon"? I have seen both terms being used but I don't know whether they referred to the same tendon.

For example,
Johnson, Anthony E., Robert R. Granville, and Thomas M. DeBerardino. "Avulsion of the common hamstring tendon origin in an active duty airman." Military medicine 168, no. 1 (2003): 40-42. (https://doi.org/10.1093/milmed/168.1.40) uses the term "common hamstring tendon" but I don't see any difference with the "hamstring tendon" when reading the article.

Since there are more than one hamstring tendons, I would have guessed that "common hamstring tendon" specifies which hamstring tendon is being referred to, but I haven't seen the term of any anatomy diagram yet.

The hamstring tendons that I am aware of are:

• Lateral hamstring tendon of the biceps femoris (long head) (2 tendons: distal+proximal)


• Lateral hamstring tendon of the biceps femoris (short head) (2 tendons: distal+proximal)


• Medial hamstring tendon of the semitendinosus (2 tendons: distal+proximal)


• Medial hamstring tendon of the semimembranosus (2 tendons: distal+proximal)

(image source)

(image source)

In the paper you cite, they are talking about the tendon on the other side from the ones you've listed, at the "origin from the ischial tuberosity" (quoted from the abstract).

In this case, the word "common" is used in the dictionary sense of "same" rather than "ordinary": it's "common" because it's the same origin for the different muscles that make up the hamstrings. This terminology is used elsewhere in anatomy, too, for example the common carotid artery splits into the internal carotid and external carotid; the "common" portion is the larger proximal vessel.

Expanding on Bryan Krause's answer:

From 1:

The semitendinosus (ST), long head of the biceps femoris (long head, lhBF) and semimembranosus (SM) muscles
originate from the ischial tuberosity (Fig. 1a, b). The ST and
lhBF have a common origin on the posteromedial aspect of
the ischial tuberosity, over its top. Tendons of the ST and
lhBF are conjoined at a distance of 9.1–10 cm [35, 37, 58,
81]. The SM origin is separate from the previous one and
it is located anterolaterally from the ST/lhBF attachment.
Fibres of the proximal SM attachment are twisted before
forming a proper tendon (Figs. 2a, b, 3a, b, 4).

A majority of authors agree with the presence of a conjoined tendon of the ST/lhBF, but the precise description of
its attachment area varies amongst authors. Most of authors
observed the attachment on the posteromedial aspect of the
ischial tuberosity as in our dissection [61, 68, 82], whereas
others indicated it to be directly medial [10, 58] or lateral on
the ischial tuberosity [35, 60]. Consequently, the SM attachment is also described in different ways: on the anterolateral
aspect of the ischial tuberosity as in our dissection [61, 68,
82, 84], but also anteriorly [35] or purely lateral [58]

Subsequently the "common hamstring tendon" = the conjoined ST tendon + lhBF tendon, but doesn't include the semimembranosus tendon as its attachment is distinct from the attachment of the conjoined ST tendon + lhBF tendon.

To visualize the attachments of the conjoined ST tendon + lhBF tendon, and the semimembranosus tendon to the ischial tuberosity:

Fig. 2 a, b Posterolateral view of the area of the proximal attachment
of the hamstring muscles (right lower extremity). (1) Area of
the attachment of the conjoined tendon of the semitendinosus and the
long head of the biceps femoris; (2) the proximal attachment area of
the conjoined tendon; (3) conjoined tendon of the semitendinosus and
the long head of the biceps femoris—cut and rotated 180°; (4) proximal
tendon of the semimembranosus muscle; (5) area of the attachment
of the semimembranosus muscle; arrowheads—shape of the
semimembranosus attachment

(image source: 1)

References:

• 1 Stępień, Karolina, Robert Śmigielski, Caroline Mouton, Bogdan Ciszek, Martin Engelhardt, and Romain Seil. "Anatomy of proximal attachment, course, and innervation of hamstring muscles: a pictorial essay." Knee Surgery, Sports Traumatology, Arthroscopy (2018): 1-12. (https://doi.org/10.1007/s00167-018-5265-z)